Chapter 6 Resource Masters€¦ · iv Teacher’s Guide to Using the Chapter 6 Resource Masters The...

Chapter 6 Resource Masters

Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13

Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4

Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1

Spanish Version

Homework Practice Workbook 0-07-890853-1 978-0-07-890853-8

Answers for Workbooks The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the English workbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9)These masters contain a Spanish version of Chapter 6 Test Form 2A and Form 2C.

Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.

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ISBN13: 978-0-07-890515-5

ISBN10: 0-07-890515-X

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1 2 3 4 5 6 7 8 9 10 024 14 13 12 11 10 09 08

iii

Teacher’s Guide to Using the Chapter 6

Resource Masters .........................................iv

Chapter ResourcesStudent-Built Glossary ....................................... 1

Anticipation Guide (English) .............................. 3

Anticipation Guide (Spanish) ............................. 4

Lesson 6-1Angles of Polygons

Study Guide and Intervention ............................ 5

Skills Practice .................................................... 7

Practice .............................................................. 8

Word Problem Practice ..................................... 9

Enrichment ...................................................... 10

Lesson 6-2Parallelograms

Study Guide and Intervention .......................... 11

Skills Practice .................................................. 13

Practice ............................................................ 14

Word Problem Practice ................................... 15

Enrichment ...................................................... 16

Lesson 6-3Tests for Parallelograms


Skills Practice .................................................. 19

Practice ............................................................ 20


Enrichment ...................................................... 22

Lesson 6-4Rectangles


Skills Practice .................................................. 25

Practice ............................................................ 26


Enrichment ...................................................... 28

TI-Nspire Activity ............................................. 29

Geometer’s Sketchpad Activity ....................... 30

Lesson 6-5Rhombi and Squares


Skills Practice .................................................. 33

Practice ............................................................ 34


Enrichment ...................................................... 36

Lesson 6-6Trapezoids and Kites


Skills Practice .................................................. 39

Practice ............................................................ 40


Enrichment ...................................................... 42

AssessmentStudent Recording Sheet ................................ 43

Rubric for Extended Response ....................... 44

Chapter 6 Quizzes 1 and 2 ............................. 45

Chapter 6 Quizzes 3 and 4 ............................. 46

Chapter 6 Mid-Chapter Test ............................ 47

Chapter 6 Vocabulary Test ............................. 48

Chapter 6 Test, Form 1 ................................... 49

Chapter 6 Test, Form 2A ................................. 51

Chapter 6 Test, Form 2B ................................. 53

Chapter 6 Test, Form 2C ................................ 55

Chapter 6 Test, Form 2D ................................ 57

Chapter 6 Test, Form 3 ................................... 59

Chapter 6 Extended-Response Test ............... 61

Standardized Test Practice ............................. 62

Unit 2 Test ....................................................... 65

Answers ......................................... A24–A34

ContentsC

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iv

Teacher’s Guide to Using the

Chapter 6 Resource MastersThe Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing on the

TeacherWorks PlusTM CD-ROM.

Chapter Resources

Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to recording definitions and/or examples for each term. You may suggest that student highlight or star the terms with which they are not familiar. Give to students before beginning Lesson 6-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This mas-ter presented in both English and Spanish is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their percep-tions have changed.

Lesson Resources

Study Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises sec-tion of the Student Edition and includes word problems. Use as an additional prac-tice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the math-ematics they are learning. They are written for use with all levels of students.

Graphing Calculator or Spreadsheet

Activities These activities present ways in which technology can be used with the con-cepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presenta-tion.

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Assessment Options

The assessment masters in the Chapter 6

Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assess-ment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test pro-vides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choice questions and is intended for use with below grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Samples answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

• Full-size answer keys are provided for the assessment masters.

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NAME DATE PERIOD

Chapter 6 1 Glencoe Geometry

Student-Built Glossary

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6.

As you study the chapter, complete each term’s definition or description.

Remember to add the page number where you found the term. Add these pages to

your Geometry Study Notebook to review vocabulary at the end of the chapter.

6

Vocabulary TermFound

on PageDefinition/Description/Example

base

diagonal

isosceles trapezoid

legs

midsegment of a trapezoid

(continued on the next page)

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Student-Built Glossary (continued)6

Vocabulary TermFound

on PageDefinition/Description/Example

parallelogram

rectangle

rhombus

square

trapezoid

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Anticipation Guide

Quadrilaterals

Before you begin Chapter 6

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 6

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

6

STEP 1A, D, or NS

StatementSTEP 2A or D

1. A triangle has no diagonals.

2. A diagonal of a polygon is a segment joining the midpoints of two sides of the polygon.

3. The sum of the measures of the angles in a polygon can be determined by subtracting 2 from the number of sides and multiplying the result by 180.

4. For a quadrilateral to be a parallelogram it must have two pairs of parallel sides.

5. The diagonals of a parallelogram are congruent.

6. If you know that one pair of opposite sides of a quadrilateral is both parallel and congruent, then you know the quadrilateral is a parallelogram.

7. If a quadrilateral is a rectangle, then all four angles are congruent.

8. The diagonals of a rhombus are congruent.

9. The properties of a rhombus are not true for a square.

10. A trapezoid has only one pair of parallel sides.

11. The median of a trapezoid is perpendicular to the bases.

12. An isosceles trapezoid has exactly one pair of congruent sides.

Step 2

Step 1

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NOMBRE FECHA PERÍODO

Antes de comenzar el Capítulo 6

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a).

Después de completar el Capítulo 6

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

Ejercicios preparatorios

Cuadriláteros

6

Paso 1

PASO 1A, D o NS

EnunciadoPASO 2A o D

1. Un triángulo no tiene diagonales.

2. La diagonal de un polígono es un segmento que une los puntos medios de dos lados del polígono.

3. La suma de las medidas de los ángulos de un polígono puede determinarse restando 2 del número de lados y multiplicando el resultado por 180.

4. Para que un cuadrilátero sea un paralelogramo debe tener dos pares de lados paralelos.

5. Las diagonales de un paralelogramo son congruentes.

6. Si sabes que un par de lados opuestos de un cuadrilátero son paralelos y congruentes, entonces sabes que el cuadrilátero es un paralelogramo.

7. Si un cuadrilátero es un rectángulo, entonces todos los cuatro lados son congruentes.

8. Las diagonales de un rombo son congruentes.

9. Las propiedades de un rombo no son verdaderas para un cuadrado.

10. Un trapecio sólo tiene dos lados paralelos.

11. La mediana de un trapecio es perpendicular a las bases.

12. Un trapecio isósceles tiene exactamente un par de lados congruentes.

Paso 2

Capítulo 6 4 Geometría de Glencoe

Less

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Study Guide and Intervention

Angles of Polygons

Polygon Interior Angles Sum The segments that connect the nonconsecutive vertices of a polygon are called diagonals. Drawing all of the diagonals from one vertex of an n-gon separates the polygon into n - 2 triangles. The sum of the measures of the interior angles of the polygon can be found by adding the measures of the interior angles of those n - 2 triangles.

A convex polygon has

13 sides. Find the sum of the measures

of the interior angles.

(n - 2) 180 = (13 - 2) 180

= 11 180

= 1980

The measure of an

interior angle of a regular polygon is

120. Find the number of sides.

The number of sides is n, so the sum of the measures of the interior angles is 120n.

120n = (n - 2) 180

120n = 180n - 360

-60n = -360

n = 6

Exercises

Find the sum of the measures of the interior angles of each convex polygon.

1. decagon 2. 16-gon 3. 30-gon

4. octagon 5. 12-gon 6. 35-gon

The measure of an interior angle of a regular polygon is given. Find the number

of sides in the polygon.

7. 150 8. 160 9. 175

10. 165 11. 144 12. 135

13. Find the value of x.

6 -1

7x°

(4x + 10)°

(4x + 5)°

(6x + 10)°

(5x - 5)°

A B

C

D

E

Polygon Interior Angle

Sum TheoremThe sum of the interior angle measures of an n-sided convex polygon is (n - 2) · 180.

Example 1 Example 2

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Study Guide and Intervention (continued)

Angles of Polygons

Polygon Exterior Angles Sum There is a simple relationship among the exterior angles of a convex polygon.

Find the sum of the measures of the exterior angles, one at

each vertex, of a convex 27-gon.

For any convex polygon, the sum of the measures of its exterior angles, one at each

vertex, is 360.

Find the measure of each exterior angle of

regular hexagon ABCDEF.

The sum of the measures of the exterior angles is 360 and a hexagon

has 6 angles. If n is the measure of each exterior angle, then

6n = 360

n = 60

The measure of each exterior angle of a regular hexagon is 60.

Exercises

Find the sum of the measures of the exterior angles of each convex polygon.

1. decagon 2. 16-gon 3. 36-gon

Find the measure of each exterior angle for each regular polygon.

4. 12-gon 5. hexagon 6. 20-gon

7. 40-gon 8. heptagon 9. 12-gon

10. 24-gon 11. dodecagon 12. octagon

AB

C

DE

F

6 -1

Polygon Exterior Angle

Sum Theorem

The sum of the exterior angle measures of a convex polygon, one angle

at each vertex, is 360.

Example 1

Example 2

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Lesso

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-1


Skills Practice

Angles of Polygons


1. nonagon 2. heptagon 3. decagon

The measure of an interior angle of a regular polygon is given. Find the

number of sides in the polygon.

4. 108 5. 120 6. 150

Find the measure of each interior angle.

7. 8.

9. 10.

Find the measures of each interior angle of each regular polygon.

11. quadrilateral 12. pentagon 13. dodecagon

Find the measures of each exterior angle of each regular polygon.

14. octagon 15. nonagon 16. 12-gon

6-1

x°

x°

(2x - 15)°

(2x - 15)°

A B

CD

(2x)°

(2x + 20)° (3x - 10)°

(2x - 10)°

L M

NP

(2x + 16)°

(x + 14)° (x + 14)°

(2x + 16)° (7x)° (7x)°

(4x)° (4x)°

(7x)° (7x)°

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1. 11-gon 2. 14-gon 3. 17-gon

The measure of an interior angle of a regular polygon is given. Find the

number of sides in the polygon.

4. 144 5. 156 6. 160

Find the measure of each interior angle.

7. 8.

Find the measures of an exterior angle and an interior angle given the number of

sides of each regular polygon. Round to the nearest tenth, if necessary.

9. 16 10. 24 11. 30

12. 14 13. 22 14. 40

15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. The

basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to

the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram.

Find the sum of the measures of the interior angles of one such face.

Practice

Angles of Polygons

6 -1

x°

(2x + 15)° (3x - 20)°

(x + 15)°

J K

MN

(6x - 4)°

(2x + 8)° (6x - 4)°

(2x + 8)°

Lesso

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Word Problem Practice

Angles of Polygons

La Tribuna

1. ARCHITECTURE In the Uffizi gallery in Florence, Italy, there is a room built by Buontalenti called the Tribune (La

Tribuna in Italian). This room is shaped like a regular octagon.

What angle do consecutive walls of the Tribune make with each other?

2. BOXES Jasmine is designing boxes she will use to ship her jewelry. She wants to shape the box like a regular polygon. In order for the boxes to pack tightly, she decides to use a regular polygon that has the property that the measure of its interior angles is half the measure of its exterior angles. What regular polygon should she use?

3. THEATER A theater floor plan is shown in the figure. The upper five sides are part of a regular dodecagon.

Find m∠1.

4. ARCHEOLOGY Archeologists unearthed parts of two adjacent walls of an ancient castle.

Before it was unearthed, they knew from ancient texts that the castle was shaped like a regular polygon, but nobody knew how many sides it had. Some said 6, others 8, and some even said 100. From the information in the figure, how many sides did the castle really have?

5. POLYGON PATH In Ms. Rickets’ math class, students made a “polygon path” that consists of regular polygons of 3, 4, 5, and 6 sides joined together as shown.

a. Find m∠2 and m∠5.

b. Find m∠3 and m∠4.

c. What is m∠1?

6 -1

2

1

3

4

5

stage

1

24˚

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Enrichment

Central Angles of Regular PolygonsYou have learned about the interior and exterior angles of a polygon. Regular polygons also have central angles. A central angle is measured from the center of the polygon.

The center of a polygon is the point equidistant from all of the vertices of the polygon, just as the center of a circle is the point equidistant from all of the points on the circle. The central angle is the angle drawn with the vertex at the center of the circle and the sides of angle drawn through consecutive vertices of the polygon. One of the central angles that can be drawn in this regular hexagon is ∠APB. You may remember from making circle graphs that there are 360° around the center of a circle.

1. By using logic or by drawing sketches, find the measure of the central angle of each regular polygon.

2. Make a conjecture about how the measure of a central angle of a regular polygon relates to the measures of the interior angles and exterior angles of a regular polygon.

3. CHALLENGE In obtuse △ABC, −−−

BC is the longest side. −−

AC is also a side of a 21-sided regular polygon.

−− AB is also a side of a 28-sided regular polygon. The

21-sided regular polygon and the 28-sided regular polygon have the same center point P. Find n if

−−− BC is a side of a n-sided regular polygon that has

center point P.

(Hint: Sketch a circle with center P and place points A, B, and C on the circle.)

6 -1

A

B

P

r

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Parallelograms

Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are four important properties of parallelograms.

If ABCD is a parallelogram, find the value of each variable.−−

AB and −−−

CD are opposite sides, so −−

AB " −−−

CD .

2a = 34

a = 17

∠A and ∠C are opposite angles, so ∠A " ∠C.

8b = 112

b = 14

Exercises

Find the value of each variable.

1. 2.

3. 4.

5. 6.

P Q

RS

BA

D C34

2a

8b°

112°

4y°

3x° 8y

88

6x°

72x

30x 150

2y

6 -2

If PQRS is a parallelogram, then

If a quadrilateral is a parallelogram,

then its opposite sides are congruent.

−−−

PQ " −−

SR and −−

PS " −−−

QR


then its opposite angles are congruent.

∠P " ∠R and ∠S " ∠Q


then its consecutive angles are

supplementary.

∠P and ∠S are supplementary; ∠S and ∠R are supplementary;

∠R and ∠Q are supplementary; ∠Q and ∠P are supplementary.

If a parallelogram has one right angle,

then it has four right angles.

If m∠P = 90, then m∠Q = 90, m∠R = 90, and m∠S = 90.

Example

3y

12

6x

60°

55° 5x°

2y°

6x° 12x°

3y°

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Parallelograms

Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.

Find the value of x and y in parallelogram ABCD.

The diagonals bisect each other, so AE = CE and DE = BE.

6x = 24 4y = 18

x = 4 y = 4.5

Exercises

Find the value of each variable.

1. 2. 3.

4. 5. 6.

COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of ABCD with the given vertices.

7. A(3, 6), B(5, 8), C(3, −2), and D(1, −4) 8. A(−4, 3), B(2, 3), C(−1, −2), and D(−7, −2)

9. PROOF Write a paragraph proof of the following.

Given: !ABCD

Prove: △AED # △BEC

A B

CD

P

A B

E

CD

6x

18

24

4y

3x

4y

812

4x

2y

28

3x

30°10

y

A B

CDE

2x°

60° 4y°

3x°2y

12

6 -2

If ABCD is a parallelogram, then

If a quadrilateral is a parallelogram, then

its diagonals bisect each other.

AP = PC and DP = PB

If a quadrilateral is a parallelogram, then each

diagonal separates the parallelogram

into two congruent triangles.

△ACD # △CAB and △ADB # △CBD

Example

x

17

4y

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Skills Practice

Parallelograms

ALGEBRA Find the value of each variable.

1. 2.

3. 4.

5. 6.

COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals

of HJKL with the given vertices.

7. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 8. H(-1, 4), J(3, 3), K(3, -2), L(-1, -1)

9. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogram

are supplementary.

6-2

x-2 y+ 1

1926

x° 44°

y°

4x-

2

3y-

1

y+5 x

+10

2b-1 b+ 3

3a- 5

2a

a+ 146a+ 4

10b+ 1

9b+ 8

(x+ 8)°

(3x)°

(y+ 9)°

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Practice

Parallelograms

ALGEBRA Find the value of each variable.

1. 2.

3. 4.

ALGEBRA Use RSTU to find each measure or value.

5. m∠RST = 6. m∠STU =

7. m∠TUR = 8. b =

COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals

of PRYZ with the given vertices.

9. P(2, 5), R(3, 3), Y(-2, -3), Z(-3, -1) 10. P(2, 3), R(1, -2), Y(-5, -7), Z(-4, -2)

11. PROOF Write a paragraph proof of the following.

Given: !PRST and !PQVU

Prove: ∠V " ∠S

12. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m∠1 is 130, find m∠2, m∠3, and m∠4.

TU

SR

B30° 23

4b - 1

25°

P R

S

U V

Q

T

1 2

34

6 -2

(4x)°

(y+10)°

(2y-40)°

4x

5y-8

3y+1

x+

6

y+3 15

x-3

12

b+1

2b

a+2

3a-4

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1. WALKWAY A walkway is made by

adjoining four parallelograms as shown.

Are the end segments a and e parallel to

each other? Explain.

2. DISTANCE Four friends live at the four

corners of a block shaped like a

parallelogram. Gracie lives 3 miles away

from Kenny. How far apart do Teresa

and Travis live from each other?

3. SOCCER Four soccer players are located

at the corners of a parallelogram. Two

of the players in opposite corners are

the goalies. In order for goalie A to be

able to see the three others, she must

be able to see a certain angle x in her

field of vision.

What angle does the other goalie have to

be able to see in order to keep an eye on

the other three players?

4. VENN DIAGRAMS Make a Venn

diagram showing the relationship

between squares, rectangles, and

parallelograms.

5. SKYSCRAPERS On vacation, Tony’s

family took a helicopter tour of the city.

The pilot said the

newest building

in the city was

the building with

this top view. He

told Tony that the

exterior angle by

the front entrance

is 72°. Tony

wanted to know

more about the building, so he drew this

diagram and used his geometry skills to

learn a few more things. The front

entrance is next to vertex B.

a. What are the measures of the four

angles of the parallelogram?

b. How many pairs of congruent

triangles are there in the figure?

What are they?


Parallelograms

A B

C D

E

72˚

Goalie BPlayer A

Goalie A

x

Player B

TeresaTravis

Gracie Kenny

a

e

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Enrichment

Diagonals of ParallelogramsIn some drawings the diagonal of a parallelogram appears to be the angle bisector of both

opposite angles. When might that be true?

1. Given: Parallelogram PQRS with diagonal −−

PR . −−

PR is an angle bisector of ∠QPS and ∠QRS.

What type of parallelogram is PQRS? Justify

your answer.

2. Given: Parallelogram WPRK with angle

bisector −−−

KD , DP = 5, and WD = 7.

Find WK and KR.

3. Refer to Exercise 2. Write a statement

about parallelogram WPRK and angle

bisector −−−

KD .

4. Given: Parallelogram ABCD with

diagonal −−−

BD and angle bisector −−

BP .

PD = 5, BP = 6, and CP = 6.

The perimeter of triangle PCD is 15.

Find AB and BC.

P

Q R

S

PD

A

BC

PD

R

K

W

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Tests for Parallelograms

Conditions for Parallelograms There are many ways to establish that a quadrilateral is a parallelogram.

Find x and y so that FGHJ is a

parallelogram.

FGHJ is a parallelogram if the lengths of the opposite sides are equal.

6x + 3 = 15 4x - 2y = 2

6x = 12 4(2) - 2y = 2

x = 2 8 - 2y = 2

-2y = -6

y = 3

Exercises

Find x and y so that the quadrilateral is a parallelogram.

1. 2.

3. 4.

5. 6.

A B

CD

E

J H

GF 6x + 3

15

4x - 2y 2

2x - 2

2y 8

12 11x°

5y° 55°

5x°5y°

25°

(x + y)°

2x°

30°

24°

6y°

3x°

9x°6y

45°18

6 -3

If: If:

both pairs of opposite sides are parallel, −−

AB || −−−

DC and −−

AD || −−−

BC ,

both pairs of opposite sides are congruent, −−

AB " −−−

DC and −−

AD " −−−

BC ,

both pairs of opposite angles are congruent, ∠ABC " ∠ADC and ∠DAB " ∠BCD,

the diagonals bisect each other, −−

AE " −−

CE and −−

DE " −−

BE ,

one pair of opposite sides is congruent and parallel, −−

AB || −−−

CD and −−

AB " −−−

CD , or −−

AD || −−−

BC and −−

AD " −−−

BC ,

then: the figure is a parallelogram. then: ABCD is a parallelogram.

Example

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Parallelograms on the Coordinate Plane On the coordinate plane, the Distance, Slope, and Midpoint Formulas can be used to test if a quadrilateral is a parallelogram.

Determine whether ABCD is a parallelogram.

The vertices are A(-2, 3), B(3, 2), C(2, -1), and D(-3, 0).

Method 1: Use the Slope Formula, m = y

2 - y

1 − x

2 - x

1

.

slope of −−−

AD = 3 - 0 −

-2 - (-3) =

3 −

1 = 3 slope of

−−− BC =

2 - (-1) −

3 - 2 =

3 −

1 = 3

slope of −−

AB = 2 - 3 −

3 - (-2) = - 1 −

5 slope of

−−− CD = -1 - 0

− 2 - (-3)

= - 1 − 5

Since opposite sides have the same slope, −−

AB || −−−

CD and −−−

AD || −−−

BC . Therefore, ABCD is a parallelogram by definition.

Method 2: Use the Distance Formula, d = √ ######### (x2 - x

1)2 + ( y

2 - y

1)2 .

AB = √ ######### (-2 - 3)2 + (3 - 2)2 = √ ### 25 + 1 or √ ## 26

CD = √ ########## (2 - (-3))2 + (-1 - 0)2 = √ ### 25 + 1 or √ ## 26

AD = √ ########## (-2 - (-3))2 + (3 - 0)2 = √ ### 1 + 9 or √ ## 10

BC = √ ######### (3 - 2)2 + (2 - (-1))2 = √ ### 1 + 9 or √ ## 10

Since both pairs of opposite sides have the same length, −−

AB $ −−−

CD and −−−

AD $ −−−

BC . Therefore, ABCD is a parallelogram by Theorem 6.9.

Exercises

Graph each quadrilateral with the given vertices. Determine whether the figure is

a parallelogram. Justify your answer with the method indicated.

1. A(0, 0), B(1, 3), C(5, 3), D(4, 0); 2. D(-1, 1), E(2, 4), F(6, 4), G(3, 1);Slope Formula Slope Formula

3. R(-1, 0), S(3, 0), T(2, -3), U(-3, -2); 4. A(-3, 2), B(-1, 4), C(2, 1), D(0, -1);Distance Formula Distance and Slope Formulas

5. S(-2, 4), T(-1, -1), U(3, -4), V(2, 1); 6. F(3, 3), G(1, 2), H(-3, 1), I(-1, 4);Distance and Slope Formulas Midpoint Formula

7. A parallelogram has vertices R(-2, -1), S(2, 1), and T(0, -3). Find all possible coordinates for the fourth vertex.

x

y

O

C

A

D

B

6 -3

Example

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Skills Practice


Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices.

Determine whether the figure is a parallelogram. Justify your answer with the

method indicated.

5. P(0, 0), Q(3, 4), S(7, 4), Y(4, 0); Slope Formula

6. S(-2, 1), R(1, 3), T(2, 0), Z(-1, -2); Distance and Slope Formulas

7. W(2, 5), R(3, 3), Y(-2, -3), N(-3, 1); Midpoint Formula

ALGEBRA Find x and y so that each quadrilateral is a parallelogram.

8. 9.

10. 11.

x + 16

y + 192y

2x - 8

2y - 11

y + 3

4x - 3

3x

(4x - 35)°

(3x + 10)°(2y - 5)°

(y + 15)°

3x - 14

x + 20

3y + 2y + 20

6 -3

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Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

3. 4.


Determine whether the figure is a parallelogram. Justify your answer with the

method indicated.

5. P(-5, 1), S(-2, 2), F(-1, -3), T(2, -2); Slope Formula

6. R(-2, 5), O(1, 3), M(-3, -4), Y(-6, -2); Distance and Slope Formulas

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

7. 8.

9. 10.

11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms?

118° 62°

118°62°

(5x + 29)°

(7x - 11)°(3y + 15)°

(5y - 9)°

3y - 5

-3x + 4

-4x - 22y + 8

-4x + 6

-6x

7y + 312y - 7

-4y - 2

x + 12

-2x + 6

y + 23

Practice


6 -3

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1. BALANCING Nikia, Madison, Angela,

and Shelby are balancing themselves

on an “X”-shaped floating object. To

balance themselves, they want to

make themselves the vertices of a

parallelogram.

In order to achieve this, do all four of

them have to be the same distance from

the center of the object? Explain.

2. COMPASSES Two compass needles

placed side by side on a table are both

2 inches long and point due north. Do

they form the sides of a parallelogram?

3. FORMATION Four jets are flying in

formation. Three of the jets are shown

in the graph. If the four jets are located

at the vertices of a parallelogram, what

are the three possible locations of the

missing jet?

4. STREET LAMPS When a coordinate

plane is placed over the Harrisville town

map, the four street lamps in the center

are located as shown. Do the four lamps

form the vertices of a parallelogram?

Explain.

5. PICTURE FRAME Aaron is making a

wooden picture frame in the shape of a

parallelogram. He has two pieces of

wood that are 3 feet long and two that

are 4 feet long.

a. If he connects the pieces of wood at

their ends to each other, in what

order must he connect them to make

a parallelogram?

b. How many different parallelograms

could he make with these four

lengths of wood?

c. Explain something Aaron might do

to specify precisely the shape of the

parallelogram.

5

5y

x

O–5

–5

5

5

y

xO

Shelby

Madison

Nikia

Angela

6 -3 Word Problem Practice


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Enrichment


By definition, a quadrilateral is a parallelogram if and only if both pairs of opposite sides

are parallel. What conditions other than both pairs of opposite sides parallel will guarantee

that a quadrilateral is a parallelogram? In this activity, several possibilities will be

investigated by drawing quadrilaterals to satisfy certain conditions. Remember that any

test that seems to work is not guaranteed to work unless it can be formally proven.

Complete.

1. Draw a quadrilateral with one pair of opposite sides congruent.

Must it be a parallelogram?

2. Draw a quadrilateral with both pairs of opposite sides congruent.


3. Draw a quadrilateral with one pair of opposite sides parallel and

the other pair of opposite sides congruent. Must it be a

parallelogram?

4. Draw a quadrilateral with one pair of opposite sides both parallel

and congruent. Must it be a parallelogram?

5. Draw a quadrilateral with one pair of opposite angles congruent.


6. Draw a quadrilateral with both pairs of opposite angles congruent.


7. Draw a quadrilateral with one pair of opposite sides parallel and

one pair of opposite angles congruent. Must it be a parallelogram?

6 -3

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Rectangles

Properties of Rectangles A rectangle is a quadrilateral with four right angles. Here are the properties of rectangles.

A rectangle has all the properties of a parallelogram.

• Opposite sides are parallel.

• Opposite angles are congruent.

• Opposite sides are congruent.

• Consecutive angles are supplementary.

• The diagonals bisect each other.

Also:

• All four angles are right angles. ∠UTS, ∠TSR, ∠SRU, and ∠RUT are right angles.

• The diagonals are congruent. −−

TR # −−−

US

T S

RU

Q

Quadrilateral RUTS

above is a rectangle. If US = 6x + 3 and

RT = 7x - 2, find x.

The diagonals of a rectangle are congruent, so US = RT.

6x + 3 = 7x - 2

3 = x - 2

5 = x

Quadrilateral RUTS

above is a rectangle. If m∠STR = 8x + 3

and m∠UTR = 16x - 9, find m∠STR.

∠UTS is a right angle, so m∠STR + m∠UTR = 90.

8x + 3 + 16x - 9 = 90

24x - 6 = 90

24x = 96

x = 4

m∠STR = 8x + 3 = 8(4) + 3 or 35

Exercises

Quadrilateral ABCD is a rectangle.

1. If AE = 36 and CE = 2x - 4, find x.

2. If BE = 6y + 2 and CE = 4y + 6, find y.

3. If BC = 24 and AD = 5y - 1, find y.

4. If m∠BEA = 62, find m∠BAC.

5. If m∠AED = 12x and m∠BEC = 10x + 20, find m∠AED.

6. If BD = 8y - 4 and AC = 7y + 3, find BD.

7. If m∠DBC = 10x and m∠ACB = 4x2- 6, find m∠ ACB.

8. If AB = 6y and BC = 8y, find BD in terms of y.

B C

DA

E

6 -4

Example 1 Example 2

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Rectangles

Prove that Parallelograms Are Rectangles The diagonals of a rectangle are congruent, and the converse is also true.

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

In the coordinate plane you can use the Distance Formula, the Slope Formula, and properties of diagonals to show that a figure is a rectangle.

Quadrilateral ABCD has vertices A(-3, 0), B(-2, 3), C(4, 1), and D(3, -2). Determine whether ABCD is a rectangle.

Method 1: Use the Slope Formula.

slope of −−

AB = 3 - 0 −

-2 - (-3) =

3 −

1 or 3 slope of

−−− AD = -2 - 0

− 3 - (-3)

= -2 −

6 or - 1 −

3

slope of −−−

CD = -2 - 1 −

3 - 4 =

-3 −

-1 or 3 slope of

−−− BC = 1 - 3

− 4 - (-2)

= -2 −

6 or - 1 −

3

Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides are perpendicular, so ABCD is a rectangle.

x

y

O

D

B

A

EC

6 -4

Example

Method 2: Use the Distance Formula.

AB = √ ########## (-3 - (-2) ) 2 + (0 - 3 ) 2 or √ ## 10 BC = √ ######## (-2 - 4 ) 2 + (3 - 1 ) 2 or √ ## 40

CD = √ ######### (4 - 3 ) 2 + (1 - (-2) ) 2 or √ ## 10 AD = √ ########## (-3 - 3 ) 2 + (0 - (-2) ) 2 or √ ## 40

Opposite sides are congruent, thus ABCD is a parallelogram.

AC = √ ######## (-3 - 4 ) 2 + (0 - 1 ) 2 or √ ## 50 BD = √ ########## (-2 - 3 ) 2 + (3 - (-2) ) 2 or √ ## 50

ABCD is a parallelogram with congruent diagonals, so ABCD is a rectangle.

Exercises

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the

indicated formula.

1. A(−3, 1), B(−3, 3), C(3, 3), D(3, 1); Distance Formula

2. A(−3, 0), B(−2, 3), C(4, 5), D(3, 2); Slope Formula

3. A(−3, 0), B(−2, 2), C(3, 0), D(2 −2); Distance Formula

4. A(−1, 0), B(0, 2), C(4, 0), D(3, −2); Distance Formula

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Skills Practice

Rectangles

ALGEBRA Quadrilateral ABCD is a rectangle.

1. If AC = 2x + 13 and DB = 4x - 1, find DB.

2. If AC = x + 3 and DB = 3x - 19, find AC.

3. If AE = 3x + 3 and EC = 5x - 15, find AC.

4. If DE = 6x - 7 and AE = 4x + 9, find DB.

5. If m∠DAC = 2x + 4 and m∠BAC = 3x + 1, find m∠BAC.

6. If m∠BDC = 7x + 1 and m∠ADB = 9x - 7, find m∠BDC.

7. If m∠ABD = 7x - 31 and m∠CDB = 4x + 5, find m∠ABD.

8. If m∠BAC = x + 3 and m∠CAD = x + 15, find m∠BAC.

9. PROOF: Write a two-column proof.

Given: RSTV is a rectangle and U is the midpoint of

−− VT .

Prove: △RUV $ △SUT

Statements Reasons


Determine whether the figure is a rectangle. Justify your answer using the

indicated formula.

10. P(-3, -2), Q(-4, 2), R(2, 4), S(3, 0); Slope Formula

11. J(-6, 3), K(0, 6), L(2, 2), M(-4, -1); Distance Formula

12. T(4, 1), U(3, -1), X(-3, 2), Y(-2, 4); Distance Formula

D C

BAE

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Practice

Rectangles

ALGEBRA Quadrilateral RSTU is a rectangle.

1. If UZ = x + 21 and ZS = 3x - 15, find US.

2. If RZ = 3x + 8 and ZS = 6x - 28, find UZ.

3. If RT = 5x + 8 and RZ = 4x + 1, find ZT.

4. If m∠SUT = 3x + 6 and m∠RUS = 5x - 4, find m∠SUT.

5. If m∠SRT = x + 9 and m∠UTR = 2x - 44, find m∠UTR.

6. If m∠RSU = x + 41 and m∠TUS = 3x + 9, find m∠RSU.

Quadrilateral GHJK is a rectangle. Find each measure if m∠1 = 37.

7. m∠2 8. m∠3

9. m∠4 10. m∠5

11. m∠6 12. m∠7


Determine whether the figure is a rectangle. Justify your answer using the

indicated formula.

13. B(-4, 3), G(-2, 4), H(1, -2), L(-1, -3); Slope Formula

14. N(-4, 5), O(6, 0), P(3, -6), Q(-7, -1); Distance Formula

15. C(0, 5), D(4, 7), E(5, 4), F(1, 2); Slope Formula

16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain.

U T

SRZ

K

21 5

436

7

J

HG

6 -4

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1. FRAMES Jalen makes the rectangular

frame shown.

In order to make sure that it is a

rectangle, Jalen measures the distances

BD and AC. How should these two

distances compare if the frame is a

rectangle?

2. BOOKSHELVES A

bookshelf consists of

two vertical planks

with five horizontal

shelves. Are each of the

four sections for books

rectangles? Explain.

3. LANDSCAPING A landscaper is

marking off the corners of a rectangular

plot of land. Three of the corners are in

place as shown.

What are the coordinates of the fourth

corner?

4. SWIMMING POOLS Antonio is

designing a swimming pool on a

coordinate grid. Is it a rectangle?

Explain.

5. PATTERNS Veronica made the pattern

shown out of 7 rectangles with four

equal sides. The side length of each

rectangle is written inside the rectangle.

a. How many rectangles can be formed

using the lines in this figure?

b. If Veronica wanted to extend her

pattern by adding another rectangle

with 4 equal sides to make a larger

rectangle, what are the possible

side lengths of rectangles that she

can add?


Rectangles

1 1

2 3

5

8

5

y

xO

5

5y

xO–5

–5

A B

D C

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Constant Perimeter

Douglas wants to fence a rectangular region of his back yard for his dog. He bought

200 feet of fence.

1. Complete the table to show the

dimensions of five different rectangular

pens that would use the entire 200 feet

of fence. Then find the area of each

rectangular pen.

2. Do all five of the rectangular pens have

the same area? If not, which one has

the larger area?

3. Write a rule for finding the dimensions of a rectangle with

the largest possible area for a given perimeter.

4. Let x represent the length of a rectangle and y the width.

Write the formula for all rectangles with a perimeter of

200. Then graph this relationship on the coordinate plane

at the right.

Julio read that a dog the size of his new pet, Bennie, should have at least 100 square feet in

his pen. Before going to the store to buy fence, Julio made a table to determine the

dimensions for Bennie’s rectangular pen.

5. Complete the table to find five possible

dimensions of a rectangular fenced area of

100 square feet.

6. Julio wants to save money by purchasing

the least number feet of fencing to enclose the

100 square feet. What will be the dimensions

of the completed pen?

7. Write a rule for finding the dimensions of a rectangle with

the least possible perimeter for a given area.

8. For length x and width y, write a formula for the area of a

rectangle with an area of 100 square feet. Then graph the

formula.

100

100

y

xO

y

xO 100

100

6-4

Perimeter Length Width Area

200 80

200 70

200 60

200 50

200 45

Area Length WidthHow much

fence to buy

100

100

100

100

100

Enrichment

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Graphing Calculator Activity

TI-Nspire: Exploring Rectangles

A quadrilateral with four right angles is a rectangle. The TI-Nspire can be used to explore some of the characteristics of a rectangle. Use the following steps to draw a rectangle.

Step 1 Set up the calculator in the correct mode.

• Choose Graphs & Geometry from the Home Menu.

• From the View menu, choose 4: Hide Axis

Step 2 Draw the rectangle

• From the 8: Shapes menu choose 3: Rectangle.

• Click once to define the corner of the rectangle. Then move and click again. The side of the rectangle is now defined. Move perpendicularly to draw the rectangle. Click to anchor the shape.

Step 3 Measure the lengths of the sides of the rectangle.

• From the 7: Measurement menu choose 1: Length (Note that when you scroll over the rectangle, the value now shown is the perimeter of the rectangle.)

• Select each endpoint of a segment of the rectangle. Then click or press Enter to anchor the length of the segment in the work area.

• Repeat for the other sides of the rectangle.

Exercises

1. What appears to be true about the opposite sides of the rectangle?

2. Draw the diagonals of the rectangle using 5: Segment from the 6: Points and Lines

Menu. Click on two opposite vertices to draw the diagonal. Repeat to draw the other diagonal.

a. Measure each diagonal using the measurement tool. What do you observe?

b. What is true about the triangles formed by the sides of the rectangle and a diagonal? Justify your conclusion.

3. Press Clear three times and select Yes to clear the screen. Repeat the steps and draw another rectangle. Do the relationships that you found for the first rectangle you drew hold true for this rectangle?

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Geometer’s Sketchpad Activity

Exploring Rectangles

A quadrilateral with four right angles is a rectangle. The Geometer’s Sketchpad is a useful

tool for exploring some of the characteristics of a rectangle. Use the following steps to draw

a rectangle.

Step 1 Use the Line tool to draw a line anywhere

on the screen.

Step 2 Use the Point tool to draw a point that is

not on the line. To draw a line perpendicular

to the first line you drew, select the first line

and the point. Then choose Perpendicular

Line from the Construct menu.

Step 3 Use the Point tool to draw a point that is

not on either of the lines you have drawn.

Repeat the procedure in Step 2 to draw

lines perpendicular to the two lines you

have drawn.

A rectangle is formed by the segments whose endpoints are the points of intersection of the

lines.

Exercises

Use the measuring capabilities of The Geometer’s Sketchpad to explore

the characteristics of a rectangle.

1. What appears to be true about the opposite sides of the rectangle that you drew? Make a

conjecture and then measure each side to check your conjecture.

2. Draw the diagonals of the rectangle by using the Selection Arrow tool to choose two

opposite vertices. Then choose Segment from the Construct menu to draw the

diagonal. Repeat to draw the other diagonal.

a. Measure each diagonal. What do you observe?

b. What is true about the triangles formed by the sides of the rectangle and a

diagonal? Justify your conclusion.

3. Choose New Sketch from the File menu and follow steps 1–3 to draw another

rectangle. Do the relationships you found for the first rectangle you drew hold true for

this rectangle also?

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Rhombi and Squares

Properties of Rhombi and Squares A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also a parallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties.

A square is a parallelogram with four congruent sides and four

congruent angles. A square is both a rectangle and a rhombus;

therefore, all properties of parallelograms, rectangles, and rhombi

apply to squares.

In rhombus ABCD, m∠BAC = 32. Find the

measure of each numbered angle.

ABCD is a rhombus, so the diagonals are perpendicular and △ABE

is a right triangle. Thus m∠4 = 90 and m∠1 = 90 - 32 or 58. The

diagonals in a rhombus bisect the vertex angles, so m∠1 = m∠2.

Thus, m∠2 = 58.

A rhombus is a parallelogram, so the opposite sides are parallel. ∠BAC and ∠3 are

alternate interior angles for parallel lines, so m∠3 = 32.

Exercises

Quadrilateral ABCD is a rhombus. Find each value or measure.

1. If m∠ABD = 60, find m∠BDC. 2. If AE = 8, find AC.

3. If AB = 26 and BD = 20, find AE. 4. Find m∠CEB.

5. If m∠CBD = 58, find m∠ACB. 6. If AE = 3x - 1 and AC = 16, find x.

7. If m∠CDB = 6y and m∠ACB = 2y + 10, find y.

8. If AD = 2x + 4 and CD = 4x - 4, find x.

M O

HR

B

CA

D

E

B

32°

1 2

3

4

C

A D

EB

6 -5

The diagonals are perpendicular. −−−

MH ⊥ −−−

RO

Each diagonal bisects a pair of opposite angles. −−−

MH bisects ∠RMO and ∠RHO.

−−−

RO bisects ∠MRH and ∠MOH.

Example

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Rhombi and Squares

Conditions for Rhombi and Squares The theorems below can help you prove that a parallelogram is a rectangle, rhombus, or square.

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.

If a quadrilateral is both a rectangle and a rhombus, then it is a square.

Determine whether parallelogram ABCD with

vertices A(−3, −3), B(1, 1), C(5, −3), D(1, −7) is a rhombus, a rectangle, or a square.

AC = √ """""""""" (-3 - 5)2 + ((-3 - (-3))2 = √ "" 64 = 8

BD = √ """""""" (1-1)2 + (-7 - 1)2 = √ "" 64 = 8

The diagonals are the same length; the figure is a rectangle.

Slope of −−

AC = -3 - (-3)

− -3 - 5 =

0 −

-8 = 8 The line is horizontal.

Slope of −−−

BD = 1 - (-7)

− 1 - 1

= 8 −

0 = undefined The line is vertical.

Since a horizontal and vertical line are perpendicular, the diagonals are perpendicular. Parallelogram ABCD is a square which is also a rhombus and a rectangle.

Exercises

Given each set of vertices, determine whether !ABCD is a rhombus, rectangle, or square. List all that apply. Explain.

1. A(0, 2), B(2, 4), C(4, 2), D(2, 0) 2. A(-2, 1), B(-1, 3), C(3, 1), D(2, -1)

3. A(-2, -1), B(0, 2), C(2, -1), D(0, -4) 4. A(-3, 0), B(-1, 3), C(5, -1), D(3, -4)

5. PROOF Write a two-column proof.

Given: Parallelogram RSTU. −−

RS $ −−

ST

Prove: RSTU is a rhombus.

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Skills Practice

Rhombi and Squares

ALGEBRA Quadrilateral DKLM is a rhombus.

1. If DK = 8, find KL.

2. If m∠DML = 82 find m∠DKM.

3. If m∠KAL = 2x – 8, find x.

4. If DA = 4x and AL = 5x – 3, find DL.

5. If DA = 4x and AL = 5x – 3, find AD.

6. If DM = 5y + 2 and DK = 3y + 6, find KL.

7. PROOF Write a two-column proof.

Given: RSTU is a parallelogram. −−−RX "

−−TX "

−−SX "

−−−UX

Prove: RSTU is a rectangle.

Statements Reasons

COORDINATE GEOMETRY Given each set of vertices, determine whether QRST

is a rhombus, a rectangle, or a square. List all that apply. Explain.

8. Q(3, 5), R(3, 1), S(-1, 1), T(-1, 5)

9. Q(-5, 12), R(5, 12), S(-1, 4), T(-11, 4)

10. Q(-6, -1), R(4, -6), S(2, 5), T(-8, 10)

11. Q(2, -4), R(-6, -8), S(-10, 2), T(-2, 6)

M L

AD K

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Practice

Rhombi and Squares

PRYZ is a rhombus. If RK = 5, RY = 13 and m∠YRZ = 67, find each measure.

1. KY

2. PK

3. m∠YKZ

4. m∠PZR

MNPQ is a rhombus. If PQ = 3 √ # 2 and AP = 3, find each measure.

5. AQ

6. m∠APQ

7. m∠MNP

8. PM

COORDINATE GEOMETRY Given each set of vertices, determine whether $BEFG

is a rhombus, a rectangle, or a square. List all that apply. Explain.

9. B(-9, 1), E(2, 3), F(12, -2), G(1, -4)

10. B(1, 3), E(7, -3), F(1, -9), G(-5, -3)

11. B(-4, -5), E(1, -5), F(-2, -1), G(-7, -1)

12. TESSELLATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.

K

P

YR

Z

A

N

M Q

P

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1. TRAY RACKS A tray rack looks like a

parallelogram from the side. The levels

for the trays are evenly spaced.

What two labeled points form a rhombus

with base AA' ?

2. SLICING Charles cuts a rhombus along

both diagonals. He ends up with four

congruent triangles. Classify these

triangles as acute, obtuse, or right.

3. WINDOWS The edges of a window are

drawn in the coordinate plane.

Determine whether the window is a

square or a rhombus.

4. SQUARES Mackenzie cut a square

along its diagonals to get four congruent

right triangles. She then joined two of

them along their long sides. Show that

the resulting shape is a square.

5. DESIGN Tatianna made the design

shown. She used 32 congruent rhombi to

create the flower-like design at each

corner.

a. What are the angles of the corner

rhombi?

b. What kinds of quadrilaterals are the

dotted and checkered figures?


Rhombi and Squares

y

xO

A

B

C

D

E

F

G

H

B,

C,

D,

E,F,G,

H,

A20 inches

AB = 4 inches,

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Enrichment

Creating Pythagorean Puzzles

By drawing two squares and cutting them in a certain way, you

can make a puzzle that demonstrates the Pythagorean Theorem.

A sample puzzle is shown. You can create your own puzzle by

following the instructions below.

1. Carefully construct a square and label the length

of a side as a. Then construct a smaller square to

the right of it and label the length of a side as b,

as shown in the figure above. The bases should

be adjacent and collinear.

2. Mark a point X that is b units from the left edge

of the larger square. Then draw the segments

from the upper left corner of the larger square to

point X, and from point X to the upper right corner

of the smaller square.

3. Cut out and rearrange your five pieces to form a

larger square. Draw a diagram to show your

answer.

4. Verify that the length of each side is equal to

√ """ a2 + b2 .

a

a

b X

b

b

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Trapezoids and Kites

Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The midsegment or median of a trapezoid is the segment that connects the midpoints of the legs of the trapezoid. Its measure is equal to one-half the sum of the lengths of the bases. If the legs are congruent, the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent and the diagonals are congruent.

The vertices of ABCD are A(-3, -1), B(-1, 3), C(2, 3), and D(4, -1). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.

slope of −−

AB = 3 - (-1)

− -1 - (-3)

= 4 − 2 = 2

slope of −−−

AD = -1 - (-1)

− 4 - (-3)

= 0 −

7 = 0

slope of −−−

BC = 3 - 3 −

2 - (-1) =

0 −

3 = 0

slope of −−−

CD = -1 - 3 −

4 - 2 = -4

− 2 = -2

Exactly two sides are parallel, −−−

AD and −−−

BC , so ABCD is a trapezoid. AB = CD, so ABCD is an isosceles trapezoid.

Exercises

Find each measure.

1. m∠D 2. m∠L

125°

5

5

40°

COORDINATE GEOMETRY For each quadrilateral with the given vertices, verify that the quadrilateral is a trapezoid and determine whether the figure is an isosceles trapezoid.

3. A(−1, 1), B(3, 2), C(1,−2), D(−2, −1) 4. J(1, 3), K(3, 1), L(3, −2), M(−2, 3)

For trapezoid HJKL, M and N are the midpoints of the legs.

5. If HJ = 32 and LK = 60, find MN.

6. If HJ = 18 and MN = 28, find LK.

T

R U

base

base

Sleg

x

y

O

B

DA

C

6-6

STUR is an isosceles trapezoid.

−−

SR $ −−

TU ; ∠R $ ∠U, ∠S $ ∠T

Example

AB = √ &&&&&&&&&& (-3 - (-1))2 + (-1 - 3)2

= √ &&& 4 + 16 = √ && 20 = 2 √ & 5

CD = √ &&&&&&&&& (2 - 4)2 + (3 - (-1))2

= √ &&& 4 + 16 = √ && 20 = 2 √ & 5

L K

NM

H J

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Properties of Kites A kite is a quadrilateral with exactly two pairs of consecutive congruent sides. Unlike a parallelogram, the opposite sides of a kite are not congruent or parallel.

The diagonals of a kite are perpendicular.

For kite RMNP, −−−

MP ⊥ −−−

RN

In a kite, exactly one pair of opposite angles is congruent.

For kite RMNP, ∠M $ ∠P

If WXYZ is a kite, find m∠Z.

The measures of ∠Y and ∠W are not congruent, so ∠X $ ∠Z.

m∠X + m∠Y + m∠Z + m∠W = 360

m∠X + 60 + m∠Z + 80 = 360

m∠X + m∠Z = 220

m∠X = 110, m∠Z = 110

If ABCD is a kite, find BC.

The diagonals of a kite are perpendicular. Use the

Pythagorean Theorem to find the missing length.

BP2 + PC2 = BC2

52 + 122 = BC2

169 = BC2

13 = BC

Exercises

If GHJK is a kite, find each measure.

1. Find m∠JRK.

2. If RJ = 3 and RK = 10, find JK.

3. If m∠GHJ = 90 and m∠GKJ = 110, find m∠HGK.

4. If HJ = 7, find HG.

5. If HG = 7 and GR = 5, find HR.

6. If m∠GHJ = 52 and m∠GKJ = 95, find m∠HGK.



6-6

Example 1

Example 2

80°

60°

5

5

4

P

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ALGEBRA Find each measure.

1. m∠S 2. m∠M

63°

14 14

142°

21 21

3. m∠D 4. RH

36° 70°20

12

12

ALGEBRA For trapezoid HJKL, T and S are midpoints of the legs.

5. If HJ = 14 and LK = 42, find TS.

6. If LK = 19 and TS = 15, find HJ.

7. If HJ = 7 and TS = 10, find LK.

8. If KL = 17 and JH = 9, find ST.

COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0), G(8, −5), H(−4, 4).

9. Verify that EFGH is a trapezoid.

10. Determine whether EFGH is an isosceles trapezoid. Explain.

6-6 Skills Practice


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Practice


Find each measure.

1. m∠T 2. m∠Y

60°

68°

7

7

3. m∠Q 4. BC

110° 48°

11 4

7

7

ALGEBRA For trapezoid FEDC, V and Y are midpoints of the legs.

5. If FE = 18 and VY = 28, find CD.

6. If m∠F = 140 and m∠E = 125, find m∠D.

COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R(−3, −3), S(5, 1), T(10, −2), U(−4, −9).

7. Verify that RSTU is a trapezoid.

8. Determine whether RSTU is an isosceles trapezoid. Explain.

9. CONSTRUCTION A set of stairs leading to the entrance of a building is designed in the

shape of an isosceles trapezoid with the longer base at the bottom of the stairs and the

shorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14 feet

wide, find the width of the stairs halfway to the top.

10. DESK TOPS A carpenter needs to replace several trapezoid-shaped desktops in a

classroom. The carpenter knows the lengths of both bases of the desktop. What other

measurements, if any, does the carpenter need?

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DC

V Y

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1. PERSPECTIVE Artists use different

techniques to make things appear to be

3-dimensional when drawing in two

dimensions. Kevin drew the walls of a

room. In real life, all of the walls are

rectangles. In what shape did he draw

the side walls to make them appear

3-dimensional?

2. PLAZA In order to give the feeling of

spaciousness, an architect decides to

make a plaza in the shape of a kite.

Three of the four corners of the plaza

are shown on the coordinate plane. If

the fourth corner is in the first

quadrant, what are its coordinates?

y

x

3. AIRPORTS A simplified drawing of

the reef runway complex at Honolulu

International Airport is shown below.

How many trapezoids are there in this

image?

4. LIGHTING A light outside a room shines

through the door and illuminates a

trapezoidal region ABCD on the floor.

A B

D C

Under what circumstances would

trapezoid ABCD be isosceles?

5. RISERS A riser is designed to elevate

a speaker. The riser consists of

4 trapezoidal sections that can be

stacked one on top of the other to

produce trapezoids of varying heights.

20 feet

10 feet

All of the stages have the same height.

If all four stages are used, the width of

the top of the riser is 10 feet.

a. If only the bottom two stages are

used, what is the width of the top

of the resulting riser?

b. What would be the width of the riser

if the bottom three stages are used?

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Enrichment

Quadrilaterals in Construction

Quadrilaterals are often used in construction work.

1. The diagram at the right represents a roof frame and shows many quadrilaterals. Find the following shapes in the diagram and shade in their edges.

a. isosceles triangle

b. scalene triangle

c. rectangle

d. rhombus

e. trapezoid (not isosceles)

f. isosceles trapezoid

2. The figure at the right represents a window. The wooden part between the panes of glass is 3 inches wide. The frame around the outer edge is 9 inches wide. The outside measurements of the frame are 60 inches by 81 inches. The height of the top and bottom panes is the same. The top three panes are the same size.

a. How wide is the bottom pane of glass?

b. How wide is each top pane of glass?

c. How high is each pane of glass?

3. Each edge of this box has been reinforced with a piece of tape. The box is 10 inches high, 20 inches wide, and 12 inches deep. What is the length of the tape that has been used?

jack rafters

hip rafter

common rafters

Roof Frame

ridge boardvalley rafter

plate

60 in.

81 in.

9 in.

3 in.

3 in.

9 in.

9 in.

10 in.

12 in.

20 in.

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Assessment


SCORE 6

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Multiple Choice

Student Recording Sheet

Use this recording sheet with pages 452–453 of the Student Edition.

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing

each number or symbol in a box. Then fill in the corresponding circle for that

number or symbol.

8. (grid in)

9.

10.

11.

12. (grid in)

13. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Extended Response

Short Response/Gridded Response

Record your answers for Question 14 on the back of this paper.

8. 12.

9

8

7

6

5

4

3

2

1

0

9

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6

General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show how

he or she arrived at the answer, the student will be awarded only 1 credit. All extended-

response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for all

parts of the question. For example, if a question has three parts, the correct response to

one or two parts of the question that required work to be shown is not considered a fully

correct response.

• Students who use trial and error to solve a problem must show their method. Merely

showing that the answer checks or is correct is not considered a complete response for

full credit.

Exercise 14 Rubric

Score Speci! c Criteria

4

Students correctly determine the quadrilateral in part a is a parallelogram since

opposite sides are congruent. Students correctly determine the quadrilateral in part b

does not contain sufficient information to prove it is a parallelogram. The two horizontal

sides must also be congruent. Students correctly determine that the quadrilateral in

part c is a parallelogram since both pairs of opposite angles are congruent.

3 A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0An incorrect solution indicating no mathematical understanding of the concept or task,

or no solution is given.

Rubric for Scoring Extended Response

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Assessm

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SCORE

1. Find the sum of the measures of the interior angles of a convex 70-gon.

2. The measure of each interior angle of a regular polygon is 172. Find the number of sides in the polygon.

3. The measure of each exterior angle of a regular polygon is 18. Find the number of sides in the polygon.

4. Given parallelogram ABCD with C(5, 4), find the coordinates of A if the diagonals

−− AC and

−−− BD intersect at (2, 7).

5. MULTIPLE CHOICE Find m∠1 in parallelogram ABCD.

A 64 C 46

B 58 D 36

1.

2.

3.

4.

5.

1. Determine whether this quadrilateral is a parallelogram. Justify your answer.

For Questions 2–4, write true or false.

2. A quadrilateral with two pairs of parallel sides is always a parallelogram.

3. The diagonals of a parallelogram are always perpendicular.

4. The slope of −−

AB and −−−

CD is 3 −

5 and the slope of

−−− BC and

−−− AD is

- 5 −

3 . ABCD is a parallelogram.

5. Refer to parallelogram ABCD. If AB = 8 cm, what is the perimeter of the parallelogram?

1.

2.

3.

4.

5.

Chapter 6 Quiz 1(Lessons 6-1 and 6-2)

Chapter 6 Quiz 2(Lesson 6-3)

6

6

A

D

B

C

6 cm


110°

46°1

D C

BA

SCORE

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NAME DATE PERIOD


SCORE

1. MULTIPLE CHOICE RSTV is a rhombus. Which of the following statements is NOT true?

A −−

RV " −−

TS

B −−

RV ⊥ −−

TS

C −−

RS ‖ −−

TV

D ∠R " ∠T

For Questions 2 and 3, refer to trapezoid MNPQ.

2. Find m∠M.

3. Find m∠Q.

4. True or false. A quadrilateral that is a rectangle and a rhombus is a square.

5. &ABCD has vertices A(4, 0), B(0, 4), C(−4, 0), and D(0, −4). Determine whether ABCD is a rectangle, rhombus, or square. List all that apply.

1.

2.

3.

4.

5.

For Questions 1 and 2, refer to kite DEFC.

1. If m∠DCF = 34 and m∠DEF = 90,

find m∠CDE.

2. If DR = 5 and RE = 5, find FE.

For Questions 3 and 4, refer to trapezoid NPQM

where X and Y are midpoints of the sides.

3. If MQ = 15 and XY = 10, find NP.

4. If NP = 13 and MQ = 18, find XY.

5. If CDEF is a trapezoid with vertices C(0, 2), D(2, 4), E(7, 3), and F(1, −3), how can you prove that it is an isosceles trapezoid?

1.

2.

3.

4.

5.

6 Chapter 6 Quiz 3(Lessons 6-4 and 6-5)

Chapter 6 Quiz 4(Lesson 6-6)

(2x + 3)°

(x + 9)°

6 SCORE

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Assessment


6 SCORE

1. Find the measure of each exterior angle of a regular 56-gon. Round to the nearest tenth.

A 3.2 B 6.4 C 173.6 D 9720

2. Given BE = 2x + 6 and ED = 5x - 12 in parallelogram ABCD, find BD.

F 6 H 18

G 12 J 36

3. If the slope of −−−

PQ is 2 − 3 and the slope of

−−− QR is - 1 −

2 , find the slope of

−− SR so that

PQRS is a parallelogram.

A 2 −

3 B

3 −

2 C -

1 −

2 D 2

4. Find m∠W in parallelogram RSTW.

F 17 H 55

G 33 J 125


A 172.5 B 360 C 8280 D 8640

D

ECB

A

6.

7.

8.

9.

10.

1.

2.

3.

4.

5.

Part II

6. Find x.

7. ABCD is a parallelogram with m∠A = 138. Find m∠B.

8. Determine whether ABCD is a parallelogram if AB = 6, BC = 12, CD = 6, and DA = 12. Justify your answer.

9. In parallelogram MLKJ, find m∠MLK and m∠LKJ.

10. XYWZ is a quadrilateral with vertices W(1, −4), X(-4, 2), Y(1, −1), and Z(−2, −3). Determine if the quadrilateral is a parallelogram. Use slope to justify your answer.

Part I Write the letter for the correct answer in the blank at the right of each question.

Chapter 6 Mid-Chapter Test(Lessons 6-1 through 6-3)

(3x + 2)°(2x - 12)°

(x - 10)°(x + 30)°

(2x - 1)°

(x + 16)°

18°

12°

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6 SCORE

Choose from the terms above to complete each sentence.

1. A quadrilateral with only one pair of opposite sides parallel and the other pair of opposite sides congruent is a(n) .

2. A quadrilateral with two pairs of opposite sides parallel is a(n)

.

3. A quadrilateral with only one pair of opposite sides parallel is a(n) .

4. A quadrilateral that is both a rectangle and a rhombus is a(n) .

5. A quadrilateral with four congruent sides is a(n) .

Write whether each sentence is true or false. If false,

replace the underlined word or number to make a true

sentence.

6. A quadrilateral with four right angles is a .

7. A quadrilateral with two pairs of congruent consecutive sides is a .

Choose the correct term to complete each sentence.

8. Segments that join opposite vertices in a quadrilateral are called (medians, diagonals).

9. The segment joining the midpoints of the nonparallel sides of a trapezoid is called the (median, diagonal).

Define each term in your own words.

10. base angles of an isosceles trapezoid

11. legs of a trapezoid

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

base

base angle

diagonal

isosceles trapezoid

legs

midsegment of a trapezoid

parallelogram

rectangle

rhombus

square

trapezoid

Chapter 6 Vocabulary Test

?

?

?

?

trapezoid

kite

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Assessment


6 SCORE

Write the letter for the correct answer in the blank at the right of each question.


A 5400 B 5040 C 360 D 168

2. Find the sum of the measures of the exterior angles of a convex 21-gon.

F 21 G 180 H 360 J 3420

3. If the measure of each interior angle of a regular polygon is 108, find the measure of each exterior angle.

A 18 B 72 C 90 D 108

4. For parallelogram ABCD, find the value of x.

F 4 H 16

G 10.25 J 21.5

5. Which of the following is a property of a parallelogram?

A The diagonals are congruent. C The diagonals are perpendicular.

B The diagonals bisect the angles. D The diagonals bisect each other.

6. Find the values of x and y so that ABCD will be a parallelogram.

F x = 6, y = 42

G x = 6, y = 22

H x = 20, y = 42

J x = 20, y = 22

7. Find the value of x so that this quadrilateral is a parallelogram.

A 44 C 90

B 46 D 134

8. Parallelogram ABCD has vertices A(0, 0), B(2, 4), and C(10, 4). Find the coordinates of D.

F D(8, 0) G D(10, 0) H D(0, 4) J D(10, 8)

9. Which of the following is a property of all rectangles?

A four congruent sides C diagonals are perpendicular

B diagonals bisect the angles D four right angles

10. ABCD is a rectangle with diagonals −−

AC and −−−

BD . If AC = 2x + 10 and

BD = 56, find the value of x.

F 23 G 33 H 78 J 122

11. ABCD is a rectangle with B(-5, 0), C(7, 0) and D(7, 3). Find the coordinates of A.

A A(-5, 7) B A(3, 5) C A(-5, 3) D A(7, -3)

x° 134°

134° 46°

(4x)°

(y - 10)°32°

24°

5x - 12

3x + 20

A D

CB

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 6 Test, Form 1

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6

12. For rhombus ABCD, find m∠1.

F 45 H 90

G 60 J 120

13. Find m∠PRS in square PQRS.

A 30 C 60

B 45 D 90

14. Choose a pair of base angles of trapezoid ABCD.

F ∠A, ∠C H ∠A, ∠D

G ∠B, ∠D J ∠D, ∠C

15. In trapezoid DEFG, find m∠D.

A 44 C 108

B 72 D 136

16. The hood of Olivia’s car is the shape of a trapezoid. The base bordering the

windshield measures 30 inches and the base at the front of the car measures

24 inches. What is the width of the median of the hood?

F 25 in. G 27 in. H 28 in. J 29 in.

17. The length of one base of a trapezoid is 44, the median is 36, and the other

base is 2x + 10. Find the value of x.

A 9 B 17 C 21 D 40

18. Given trapezoid ABCD with median −−

EF , which

of the following is true?

F EF = 1 − 2 AD H AB = EF

G AE = FD J EF = BC + AD −

2

19. PQRS is a kite. Find m∠S. A 100 C 200

B 160 D 360

20. JKLM is a kite, find JM.

F √ $$ 29 H √ $$ 13

G √ $$ 89 J 11

Bonus Find x and m∠WYZ in

rhombus XYZW.

X Y

ZW

(3x + 20)°

(x 2)°

DA

B C

FE

D G

E F136°

72°

P S

Q R

1

A D

CB

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 6 Test, Form 1 (continued)

D C

A B

124° 36°

8 2

5

5

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Assessment


6 SCORE



A 8100 B 7740 C 360 D 172


F 30 H 102

G 66 J 138


A 39 B 90 C 180 D 360

4. Which of the following is a property of a parallelogram?

F Each pair of opposite sides is congruent.

G Only one pair of opposite angles is congruent.

H Each pair of opposite angles is supplementary.

J There are four right angles.

5. For parallelogram ABCD, find m∠1.

A 60 C 36

B 54 D 18

6. ABCD is a parallelogram with diagonals intersecting at E. If AE = 3x + 12 and EC = 27, find the value of x.

F 5 G 17 H 27 J 47

7. Find the values of x and y so that this quadrilateral is a parallelogram.

A x = 13, y = 24 C x = 7, y = 24

B x = 13, y = 6 D x = 7, y = 6

8. Find the value of x so that this quadrilateral is a parallelogram.

F 12 H 36

G 24 J 132

9. Parallelogram ABCD has vertices A(8, 2), B(6, -4), and C(-5, -4). Find the coordinates of D.

A D(-5, 2) B D(-3, 2) C D(-2, 2) D D(-4, 8)

10. ABCD is a rectangle. If AC = 5x + 2 and BD = x + 22, find the value of x.

F 5 G 6 H 11 J 26

11. Which of the following is true for all rectangles?

A The diagonals are perpendicular.

B The diagonals bisect the angles.

C The consecutive sides are congruent.

D The consecutive sides are perpendicular.

(2x + 60)°

(4x - 12)°

(2x + 30)°

(5x - 9)°

(y - 12)°(1–2y)°

120°

36°

1

A D

CB

(2x + 10)°

(x + 40)°

(x - 20)°x°

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 6 Test, Form 2A

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12. ABCD is a rectangle with B(-4, 6), C(-4, 2), and D(10, 2). Find the coordinates of A.

F A(6, 4) G A(10, 4) H A(2, 6) J A(10, 6)

13. For rhombus GHJK, find m∠1.

A 22 C 68

B 44 D 90

14. The diagonals of square ABCD intersect at E. If AE = 2x + 6 and BD = 6x - 10, find AC.

F 11 G 28 H 56 J 90

15. ABCD is an isosceles trapezoid with A(10, -1), B(8, 3), and C(-1, 3). Find the coordinates of D.

A D(-3, -1) B D(-10, -11) C D(-1, 8) D D(-3, 3)

16. For isosceles trapezoid MNOP, find m∠MNP.

F 44 H 80

G 64 J 116

17. The length of one base of a trapezoid is 19 inches and the length of the median is 16 inches. Find the length of the other base.

A 35 in. B 19 in. C 17.5 in. D 13 in.

18. Judith built a fence to surround her property. On a coordinate plane, the four

corners of the fence are located at (-16, 1), (-6, 5), (4, 1), and (-6, -3). Which of the following most accurately describes the shape of Judith’s fence?

F square H rhombus

G rectangle J trapezoid

19. For kite PQRS, find m∠S

A 248 C 112

B 68 D 124

20. ABCD is a parallelogram with coordinates A(4, 2), B(4, -1), C(-2, -1), and D(-2, 2). To prove that ABCD is a rectangle, you would plot the parallelogram on a coordinate plane and then find which of the following?

F measures of the angles H slopes of the diagonals

G lengths of the diagonals J midpoints of the diagonals

Bonus Find the possible value(s) of x in rectangle JKLM.

J K

LM

x 2 + 8

3x + 36

P

ON

M36°

44°

22°

1G H

JK

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 6 Test, Form 2A (continued)

22°

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Assessment


6 SCORE



A 9000 B 8640 C 360 D 172.8


F 16 H 50

G 34 J 70


A 5.54 B 90 C 180 D 360

4. Which of the following is a property of all parallelograms?

F Each pair of opposite angles is congruent.

G Only one pair of opposite sides is congruent.

H Each pair of opposite angles is supplementary.

J There are four right angles.


A 19 C 52

B 38 D 56

6. ABCD is a parallelogram with diagonals intersecting at E. If AE = 4x - 8 and EC = 36, find the value of x.

F 7 G 11 H 15.5 J 38

7. Find the values of the values of x and y so that the quadrilateral is a parallelogram.

A x = 27, y = 90 C x = 13, y = 90

B x = 27, y = 40 D x = 13, y = 40

8. Find the value of x so that the quadrilateral is a parallelogram.

F 7 1 − 3 H 12

G 8 J 66

9. ABCD is a parallelogram with A(5, 4), B(-1, -2), and C(8, -2). Find the coordinates of D.

A D(-5, 4) B D(8, 2) C D(14, 4) D D(4, 1)

10. ABCD is a rectangle. If AB = 7x - 6 and CD = 5x + 30, find the value of x.

F 5 1 − 3 G 12 H 13 J 18

11. Which of the following is true for all rectangles?

A The diagonals are perpendicular.

B The consecutive angles are supplementary.

C The opposite sides are supplementary.

D The opposite angles are complementary.

6x - 6 3x + 30

(4x - 4)°

(3x + 23)°

(y - 60)°

1–3y °

38°

124°1

A D

B C

(2x - 10)°

(x + 30)°

(x + 70)° x°

(2x)°

(2x)°

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 6 Test, Form 2B

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6

12. ABCD is a rectangle with B(-7, 3), C(5, 3), and D(5, -8). Find the

coordinates of A.

F A(-8, -7) G A(-7, -8) H A(-5, -3) J A(-8, -5)

13. For rhombus GHJK, find m∠1.

A 90 C 52

B 64 D 38

14. The diagonals of square ABCD intersect at E. If AE = 3x - 4 and BD = 10x - 48, find AC.

F 90 G 52 H 26 J 10

15. ABCD is an isosceles trapezoid with A(0, -1), B(-2, 3), and D(6, -1). Find the coordinates of C.

A C(6, 1) B C(9, 4) C C(2, 3) D C(8, 3)

16. For isosceles trapezoid MNOP, find m∠MNP.

F 42 H 82

G 70 J 98

17. The length of one base of a trapezoid is 19 meters and the length of the median is 23 meters. Find the length of the other base.

A 15 m B 21 m C 27 m D 42 m

18 On a coordinate plane, the four corners of Ronald’s garden are located at (0, 2), (4, 6), (8, 2), and (4, -2). Which of the following most accurately describes the shape of Ronald’s garden?

F square H rhombus

G rectangle J trapezoid

19. For kite WXYZ, find m∠W.

A 106 C 212

B 148 D 360

20. ABCD is a parallelogram with coordinates A(4, 2), B(3, −1), C(−1, −1), andD(−1, 2). To prove that ABCD is a rhombus, you would plot the parallelogram on a coordinate plane and then find which of the following?

F measures of the angles H slopes of the diagonals

G lengths of the diagonals J midpoints of the diagonals

Bonus The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of the polygon.

28°

42°

P

ON

M

128°

1

G K

JH

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

130° 18°

Chapter 6 Test, Form 2B (continued)

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Assessment


6 SCORE

1. What is the sum of the interior angles of an octagonal box?

2. A convex pentagon has interior angles with measures (5x - 12)°, (2x + 100)°, (4x + 16)°, (6x + 15)°, and (3x + 41)°. Find the value of x.

3. If the measure of each interior angle of a regular polygon is 171, find the number of sides in the polygon.

4. In parallelogram ABCD,

m∠1 = x + 12, and m∠2 = 6x - 18.

Find m∠1.

5. Find the measure of each exterior angle of a regular 45-gon.

6. In parallelogram ABCD, m∠A = 58. Find m∠B.

7. Find the coordinates of the intersection of the diagonals of parallelogram XYZW with vertices X(2, 2), Y(3, 6), Z(10, 6), and W(9, 2).

8. Determine whether ABCD is a parallelogram. Justify your answer.

9. Determine whether the quadrilateral with vertices A(5, 7), B(1, -2), C(-6, -3), and D(2, 5) is a parallelogram. Use the slope formula.

10. For quadrilateral ABCD, the slope of −−

AB is 1 − 4 , the slope of

−−− BC

is - 2

− 3

, and the slope of −−−

CD is 1 − 4 . Find the slope of

−−− DA so that

ABCD will be a parallelogram.

11. Given rectangle ABCD, find the value of x.

12. ABCD is a parallelogram and −−

AC # −−−

BD . Determine whether ABCD is a rectangle. Justify your answer.

13. ABCD is a rhombus with diagonals intersecting at E. If

m∠ABC is three times m∠BAD, find m∠EBC.

A B

CD

(2x + 10)°

(3x - 30)°

A D

CB

12 C

D

A

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Chapter 6 Test, Form 2C

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6

14. TUVW is a square with U(10, 2), V(8, 8), and W(2, 6). Find the

coordinates of T.

15. For isosceles trapezoid MNOP, find

m∠MNQ.

16. ABCD is a quadrilateral with vertices A(8, 3), B(6, 7), C(-1, 5), and D(-6, -1). Determine whether ABCD is a trapezoid. Justify your answer.

17. The length of the median of trapezoid EFGH is 13 feet. If the bases have lengths 2x + 4 and 10x - 50, find x.

18. ABCD is a kite, If RC = 10, and BD = 48, find CD.


19. A rectangle is always a parallelogram.

20. The diagonals of a rhombus are always perpendicular.

21. The diagonals of a square always bisect each other.

22. A trapezoid always has two congruent sides.

23. The median of a trapezoid is always parallel to the bases.

24. A kite has exactly two congruent angles.

25. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rectangle.

Bonus In parallelogram ABCD, AB = 2x - 7, BC = x + 3y, CD = x + y, and AD = 2x - y - 1. Find the values of x and y.

N

Q

O

PM59°

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

Chapter 6 Test, Form 2C (continued)

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Assessment


6 SCORE

1. Bruce is building a tabletop in the shape of an octagon. Find the sum of the external angles of the tabletop.

2. A convex octagon has interior angles with measures (x + 55)°, (3x + 20)°, 4x°, (4x - 10)°, (6x - 55)°, (3x + 52)°, 3x°, and (2x + 30)°. Find the value of x.

3. If the measure of each interior angle of a regular polygon is 176 find the number of sides in the polygon.

4. In parallelogram ABCD,

m∠1 = x + 25, and m∠2 = 2x.

Find m∠2.

5. Find the measure of each exterior angle of a regular 100-gon.

6. In parallelogram ABCD, m∠A = 63. Find m∠B.

7. Find the coordinates of the intersection of the diagonals of parallelogram XYZW with vertices X(3, 0), Y(3, 8), Z(-2, 6), and W(-2, -2).

8. Determine whether this quadrilateral is a parallelogram. Justify your answer.

9. Determine whether a quadrilateral with vertices A(5, 7), B(1, -1), C(-6, -3), and D(-2, 5) is a parallelogram. Use the slope formula.

10. If the slope of −−

AB is 1 − 2 , the slope of

−−− BC is -4, and the slope of

−−− CD is 1 −

2 , find the slope of

−−− DA so that ABCD is a parallelogram.

11. For rectangle ABCD, find the value of x.

12. ABCD is a parallelogram and m∠A = 90. Determine whether ABCD is a rectangle. Justify your answer.

B C

DA

(6x + 20)°

(3x - 2)°

1

2A D

CB

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Chapter 6 Test, Form 2D

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6

13. ABCD is a rhombus with diagonals intersecting at E. If

m∠ABC is four times m∠BAD, find m∠EBC.

14. PQRS is a square with Q(-2, 8), R(5, 7), and S(4, 0). Find the coordinates of P.

15. For isosceles trapezoid MNOP, find m∠MNQ.

16. ABCD is a quadrilateral with A(8, 21), B(10, 27), C(26, 26), and D(18, 2). Determine whether ABCD is a trapezoid. Justify your answer.

17. The length of the median of trapezoid EFGH is 17 centimeters. If the bases have lengths 2x + 4 and 8x - 50, find the value of x.

18. For kite ABCD, if RA = 15, and BD = 16, find AD.


19. A parallelogram always has four right angles.

20. The diagonals of a rhombus always bisect the angles.

21. A rhombus is always a square.

22. A rectangle is always a square.

23. The diagonals of an isosceles trapezoid are always congruent.

24. The median of a trapezoid always bisects the angles.

25. The diagonals of a kite are always perpendicular.

Bonus The measure of each interior angle of a regular polygon is 24 more than 38 times the measure of each exterior angle. Find the number of sides of the polygon.

M P

ON

Q74°

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

Chapter 6 Test, Form 2D (continued)

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NAME DATE PERIOD

Assessment


6 SCORE

1. The sum of the interior angles of an animal pen is 900°. How many sides does the pen have?

2. A convex hexagon has interior angles with measures x°, (5x - 103)°, (2x + 60)°, (7x - 31)°, (6x - 6)°, and (9x - 100)°. Find the value of x and the measure of each angle.

3. Find the measure of each exterior angle of a regular 2x-gon.


5. ABCD is a parallelogram with diagonals that intersect each other at E. If AE = x2 and EC = 6x - 8, find all possible values of AC.

6. Determine whether the quadrilateral is a parallelogram. Justify your answer.

7. For quadrilateral ABCD, the slope of −−

AB is 2 − 3 and the slope

of −−−

BC is -2. Find the slopes of −−−

CD and −−−

DA so that ABCD will be a parallelogram.

8. In rectangle ABCD, find m∠1.

9. The diagonals of rhombus ABCD intersect at E. If

m∠BAE = 2 − 3 (m∠ABE), find m∠BCD.

10. The diagonals of square ABCD intersect at E. If AE = 2, find the perimeter of ABCD.

11. For isosceles trapezoid ABCD, find AE.

12. Points G and H are midpoints of −−

AF and −−−

DE inregular hexagon ABCDEF. If AB = 6 find GH.

13. The vertices of trapezoid ABCD are A(10, -1), B(6, 6), C(-2, 6), and D(-8, -1). Find the length of the median.

B C

F

A D

HG

E

A D

CB

E F60°

8

B C

DA

(3x - 16)°

1(2x + 10)°

96° 31°1

D C

BA

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Chapter 6 Test, Form 3

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6

14.

15.

16.

17.

18.

19.

20.

B:

14. Determine whether the quadrilateral ABCD with vertices A(0,−1), B(−4,−3), C(−5, 1), D(1, 7) is a kite. Justify your answer.

15. Determine whether the quadrilateral ABCD with vertices A(6, 2), B(2, 10), C(-6, 6), and D(-2, -2) is a rectangle. Justify your answer.

16. Determine whether quadrilateral ABCD with vertices A(1, 6), B(7, 6), C(2, -3), and D(-4, -3) is a parallelogram. Use the distance formula.

17. Find the value of x in kite EFGH.

For Questions 18 and 19, complete

the two-column proof by supplying

the missing information for each

corresponding location.

Given: ABCD is a parallelogram.

−−−

BQ −−−

DS , −−

PA −−−

RC

Prove: PQRS is a parallelogram.

Statements Reasons

1. ABCD is a #.

2. −−

AD −−

CB

3. −−

PA −−

RC

4. −−

PD −−

RB

5. −−

AB −−

CD

6. −−

BQ −−

DS

7. −−

AQ −−

CS

8. ∠B ∠D, ∠A ∠C

9. △QBR △SDP, △PAQ △RCS

10. −−

QP −−

RS , −−

QR −−

PS

11. PQRS is a parallelogram.

1. Given

2. (Question 18)

3. Given

4. Seg. Sub. Prop.

5. Opp. sides of a # are .

6. Given

7. Seg. Sub. Prop.

8. Opp. & of a # are .

9. SAS

10. CPCTC

11. (Question 19)

20. In isosceles trapezoid ABCD, AE = 2x + 5, EC = 3x - 12, and BD = 4x + 20. Find the value of x.

Bonus If three of the interior angles of a convex hexagon each measure 140, a fourth angle measures 84, and the measure of the fifth angle is 3 times the measure of the sixth angle, find the measure of the sixth angle.

(x + 5)°

x°

(140 - x)°

B C

EDA

A P

R

D

C

SQ

B

Chapter 6 Test, Form 3 (continued)

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NAME DATE PERIOD

Assessment


6 SCORE Chapter 6 Extended-Response Test

Demonstrate your knowledge by giving a clear, concise solution to

each problem. Be sure to include all relevant drawings and justify

your answers. You may show your solution in more than one way or

investigate beyond the requirements of the problem.

1. a. Draw a regular convex polygon and a convex polygon that is not

regular, each with the same number of sides.

b. Label the measures of each exterior angle on your figures.

c. Find the sum of the exterior angles for each figure. What conjecture can

be made?

2. Draw a rectangle. Connect the midpoints of the consecutive sides. What

type of quadrilateral is formed? How do you know?

3. Draw an example to show why one pair of opposite sides congruent and the

other pair of opposite sides parallel is not sufficient to form a

parallelogram.

4. a. Name a property that is true for a square and not always true for a

rectangle.

b. Name a property that is true for a square and not always true for a

rhombus.

c. Name a property that is true for a rectangle and not always true for a

parallelogram.

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NAME DATE PERIOD


6 SCORE

1. Find the coordinates of X if V(0.5, 5) is the midpoint of −−− UX with

U(15, 21). (Lesson 1-3)

A (-14, -11) C (0, 0)

B (7.75, 22.5) D (15.5, -5)

2. Which of the following are possible measures for vertical angles G and H? (Lesson 2-8)

F m∠G = 125 and m∠H = 55

G m∠G = 125 and m∠H = 125

H m∠G = 55 and m∠H = 45

J m∠G = 55 and m∠H = 152.5

3. Determine which lines are parallel. (Lesson 3-5)

A # $% NS ‖ # $% PT C # $% QR ‖ # $% ST

B # $% NP ‖ # $% ST D # $% NP ‖ # $$% QR

4. Find the coordinates of B, the midpoint of −− AC , if A(2a, b) and

C(0, 2b). (Lesson 4-8)

F (2a, 2b) G (a, b) H (a, 3 −

2 b) J (

3 −

2 a, b)

5. If −− RV is an angle bisector, find

m∠UVT. (Lesson 5-1)

A 10 C 68

B 34 D 136

6. Find the slope of the line that passes through points A(-7, 14) and B(5, -2). (Lesson 3-3)

F - 4 − 3 G -

3 −

4 H

3 −

4 J

4 −

3

7. Which statement ensures that quadrilateral QRST is a parallelogram? (Lesson 6-3)

A ∠Q ' ∠S C −−− QT ‖

−− RS

B −−− QR '

−− TS and −−− QR ‖

−− TS D m∠Q + m∠S = 180

Q R

ST

(2y + 14)°

(4y - 6)°

T VS

R

U

67°

113°

65°

N P

ST

RQ

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Standardized Test Practice(Chapters 1–6)

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.

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Assessment


6

8. What is the equation of the line that contains (-12, 9) and is perpendicular to the line y =

2 −

3 x + 5? (Lesson 3-4)

F y = - 3 − 2 x - 9 H y = - 2 −

3 x - 1

G y = 3 − 2 x - 1 J y = 2 −

3 x + 17

9. Which of the following theorems can be used to prove △ABC # △DEC? (Lesson 4-5)

A SSS C SAS

B AAS D ASA

10. What is the value of x? (Lesson 6-6)

F 2 H 5.5

G 4 J 7

11. For △ABC, AB = 6 and BC = 17. Which of the following is a possible length for

−−− AC ? (Lesson 5-3)

A 5 B 9 C 13 D 24

12. What is m∠T in kite STVW?

F 100 H 95

G 130 J 260

A

D

B C

E

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13. If △UVW is an isosceles triangle, −−−

UV # −−−

WU , UV = 16b - 40, VW = 6b, and WU = 10b + 2, find the value of b. (Lesson 4-1)

14. Find the sum of the measures of the interior angles for a convex heptagon. (Lesson 6-1)

13.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

14.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Standardized Test Practice (continued)

4x - 4

16

20

Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the answer

in a column box and then shading in the appropriate circle that

corresponds to that entry.

85° 15°

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6

15. A polygon has six congruent sides. Lines containing two of its sides contain points in its interior. Name the polygon by its number of sides, and then classify it as convex or concave and regular or irregular. (Lesson 1-6)

16. If −−

RT " −−−

QM and RT = 88.9 centimeters, find QM. (Lesson 2-7)

17. Which segment is the shortest segment

from D to # $$% JM ? (Lesson 5-2)

18. If △ABC " △WXY, AB = 72, BC = 65, CA = 13, XY = 7x - 12, and WX = 19y + 34, find the values of x and y. (Lesson 4-3)

19. Freda bought two bells for just over $90 before tax. State the assumption you would make to write an indirect proof to show that at least one of the bells costs more than $45. (Lesson 5-4)

20. The area of the base of a cylinder is 5 cm2 and the height of the cylinder is 8 cm. Find the volume of the cylinder. (Lesson 1-7)

21. JKLM is a kite. Complete each statement. (Lesson 6-6)

a. −−−

MJ "

b. −−−

MK ⊥

c. m∠L = m∠

D

J K L H M

15.

16.

17.

18.

19.

20.

21. a.

b.

c.

Part 3: Short Response

Instructions: Place your answers in the space provided.

6

6

Standardized Test Practice (continued)

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NAME DATE PERIOD

Assessment


6 SCORE

1. Use a protractor to classify △UVW, △UWX, and △XWY as acute, equiangular, obtuse, or right.

2. In the figure, ∠1 " ∠2. Find the measures of the numbered angles.

3. Name the corresponding congruent sides for △AFP " △STX.

4. Determine whether △ABC " △PQR given A(2, -7), B(5, 3), C(-4, 6), P(8, -1), Q(11, 9), and R(2, 12).

5. In the figure, −−

LK bisects ∠JKM and ∠KLJ " ∠KLM. Determine which theorem or postulate can be used to prove that △JKL " △MKL.

6. Triangle ABC is isosceles with AB = BC. Name a pair of congruent angles in this triangle.

7. For kite WXYZ, find m∠Z.

For Questions 8 and 9, refer to the figure.

8. Find the value of a and m∠ZWT if −−−

ZW is an altitude of △XYZ, m∠ZWT = 3a + 5, and m∠TWY = 5a + 13.

9. Determine which angle has the greatest measure: ∠YWZ, ∠WZY, or ∠ZYW.

10. Mr. Ramirez bought a stove and a dishwasher for just over $1206. State the assumption you would make to start an indirect proof to show that at least one of the appliances cost more than $603.

XW

Y

T

Z

K L

J

M

65°

70°

110°13

2

D

E F H

G

V

U

W

X

Y

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Unit 2 Test(Chapters 4 – 6)

43°95°

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6

11. Determine whether 128 feet, 136 feet, and 245 feet can be the lengths of the sides of a triangle.

12. Write an inequality to describe the possible values of x.

13. The measure of an interior angle of a regular polygon is 140. Find the number of sides in the polygon.

14. For parallelogram JKMH, find

m∠JHK, m∠HMK, and the value of x.

15. Determine whether the vertices of quadrilateral DEFG form a parallelogram given D(-3, 5), E(3, 6), F(-1, 0), and G(6, 1).

16. For rectangle WXYZ with diagonals −−−

WY and −−

XZ ,

WY = 3d + 4 and XZ = 4d - 1, find the value of d.

17. If m∠BEC = 9z + 45 in rhombus ABCD, find the value of z.

18. In trapezoid HJLK, M and N are midpoints of the legs. Find KL.

19. Prove that quadrilateral PQRS isNOT a parallelogram.

JH

K L

NM

45

28

A C

B

D

E

J

K

M

H20°

3x + 8

7x - 24

52°

2x - 512

12

14

11

14

41˚38˚

11.

12.

13.

14.

15.

16.

17.

18.

19.

x

y

-1

1

2

3

4

5

1 2 3 4-2-3-4

Unit 2 Test (continued)

Answers

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Chapter 6 A1 Glencoe Geometry

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter Resources

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exact

ly o

ne p

air

of

con

gru

en

t si

des.

Ste

p 2

Ste

p 1

A D A A D A A D D A D A

Lesson 6-1


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

5

Gle

nc

oe

Ge

om

etr

y

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

An

gle

s o

f P

oly

go

ns

Po

lyg

on

In

teri

or

An

gle

s S

um

T

he s

egm

en

ts t

hat

con

nect

th

e n

on

con

secu

tive

vert

ices

of

a p

oly

gon

are

call

ed

dia

go

na

ls.

Dra

win

g a

ll o

f th

e d

iagon

als

fr

om

on

e v

ert

ex o

f an

n-g

on

sep

ara

tes

the p

oly

gon

in

to n

- 2

tri

an

gle

s. T

he s

um

of

the m

easu

res

of

the i

nte

rior

an

gle

s of

the p

oly

gon

can

be f

ou

nd

by a

dd

ing t

he m

easu

res

of

the i

nte

rior

an

gle

s of

those

n -

2 t

rian

gle

s.

A

co

nv

ex

po

lyg

on

ha

s

13

sid

es.

Fin

d t

he

su

m o

f th

e m

ea

su

re

s

of

the

in

terio

r a

ng

les.

(n

- 2

)

180 =

(13 -

2)

180

= 1

1

180

= 1

980

T

he

me

asu

re

of

an

inte

rio

r a

ng

le o

f a

re

gu

lar p

oly

go

n i

s

12

0.

Fin

d t

he

nu

mb

er o

f sid

es.

Th

e n

um

ber

of

sid

es

is n

, so

th

e s

um

of

the

measu

res

of

the i

nte

rior

an

gle

s is

120n

.

12

0n

= (

n -

2)

180

12

0n

= 1

80n

- 3

60

-60

n =

-360

n

= 6

Exerc

ises

Fin

d t

he

su

m o

f th

e m

ea

su

re

s o

f th

e i

nte

rio

r a

ng

les o

f e

ach

co

nv

ex

po

lyg

on

.

1

. d

eca

gon

2. 16-g

on

3. 30-g

on

4. oct

agon

5. 12-g

on

6. 35-g

on

Th

e m

ea

su

re

of

an

in

terio

r a

ng

le o

f a

re

gu

lar p

oly

go

n i

s g

ive

n.

Fin

d t

he

nu

mb

er

of

sid

es i

n t

he

po

lyg

on

.

7

. 150

8. 160

9. 175

10

. 165

11

. 144

12

. 135

13. F

ind

th

e v

alu

e o

f x.

6 -1

7x°

( 4x+

10) °

( 4x+

5) ° ( 6

x+

10) °

( 5x-

5) °

AB

C

D

E

Po

lyg

on

In

teri

or

An

gle

Su

m T

he

ore

mT

he

su

m o

f th

e in

terio

r a

ng

le m

ea

su

res o

f a

n n

-sid

ed

co

nve

x p

oly

go

n is (

n -

2)

· 1

80

.

Exam

ple

1Exam

ple

2

1

440

2520

5040

1

080

1800

5940

1

2

18

72

2

4

10

8

2

0

Answers (Anticipation Guide and Lesson 6-1)

Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

6

Gle

nc

oe

Ge

om

etr

y

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

An

gle

s o

f P

oly

go

ns

Po

lyg

on

Ex

teri

or

An

gle

s S

um

T

here

is

a s

imp

le r

ela

tion

ship

am

on

g

the e

xte

rior

an

gle

s of

a c

on

vex p

oly

gon

.

F

ind

th

e s

um

of

the

me

asu

re

s o

f th

e e

xte

rio

r a

ng

les,

on

e a

t

ea

ch

ve

rte

x,

of

a c

on

ve

x 2

7-g

on

.

For

an

y c

on

vex p

oly

gon

, th

e s

um

of

the m

easu

res

of

its

exte

rior

an

gle

s, o

ne a

t each

vert

ex,

is 3

60.

F

ind

th

e m

ea

su

re

of

ea

ch

ex

terio

r a

ng

le o

f

re

gu

lar h

ex

ag

on

ABCDEF

.

Th

e s

um

of

the m

easu

res

of

the e

xte

rior

an

gle

s is

360 a

nd

a h

exagon

has

6 a

ngle

s. I

f n

is

the m

easu

re o

f each

exte

rior

an

gle

, th

en

6n

= 3

60

n

= 6

0

Th

e m

easu

re o

f each

exte

rior

an

gle

of

a r

egu

lar

hexagon

is

60.

Exerc

ises

Fin

d t

he

su

m o

f th

e m

ea

su

re

s o

f th

e e

xte

rio

r a

ng

les o

f e

ach

co

nv

ex

po

lyg

on

.

1

. d

eca

gon

2. 16-g

on

3. 36-g

on

Fin

d t

he

me

asu

re

of

ea

ch

ex

terio

r a

ng

le f

or e

ach

re

gu

lar p

oly

go

n.

4. 12-g

on

5. h

exagon

6. 20-g

on

7

. 40-g

on

8. h

ep

tagon

9. 12-g

on

10

. 24-g

on

1

1. d

od

eca

gon

1

2. oct

agon

AB

C

DE

F

6 -1

Po

lyg

on

Ex

teri

or

An

gle

Su

m T

he

ore

m

Th

e s

um

of

the

exte

rio

r a

ng

le m

ea

su

res o

f a

co

nve

x p

oly

go

n,

on

e a

ng

le

at

ea

ch

ve

rte

x,

is 3

60

.

Exam

ple

1

Exam

ple

2

3

60

360

360

3

0

60

18

9

5

1.4

3

0

1

5

30

45


NA

ME

DA

TE

PE

RIO

D

Lesson 6-1

Ch

ap

ter

6

7

Gle

nc

oe

Ge

om

etr

y

Sk

ills

Pra

ctic

e

An

gle

s o

f P

oly

go

ns

Fin

d t

he

su

m o

f th

e m

ea

su

re

s o

f th

e i

nte

rio

r a

ng

les o

f e

ach

co

nv

ex

po

lyg

on

.

1

. n

on

agon

2. h

ep

tagon

3. d

eca

gon

Th

e m

ea

su

re

of

an

in

terio

r a

ng

le o

f a

re

gu

lar p

oly

go

n i

s g

ive

n.

Fin

d t

he

nu

mb

er o

f sid

es i

n t

he

po

lyg

on

.

4. 108

5. 120

6. 150

Fin

d t

he

me

asu

re

of

ea

ch

in

terio

r a

ng

le.

7

.

8.

9

.

10

.

Fin

d t

he

me

asu

re

s o

f e

ach

in

terio

r a

ng

le o

f e

ach

re

gu

lar p

oly

go

n.

11

. qu

ad

rila

tera

l 1

2. p

en

tagon

1

3. d

od

eca

gon

Fin

d t

he

me

asu

re

s o

f e

ach

ex

terio

r a

ng

le o

f e

ach

re

gu

lar p

oly

go

n.

14

. oct

agon

15

. n

on

agon

1

6. 12-g

on

6-1

x°

x°

( 2x-

15) °

( 2x-

15) °

AB

CD

( 2x) °

( 2x+

20) °

( 3x-

10) °

( 2x-

10) °

LM

NP

(2x+

16)°

(x+

14)°

(x+

14

)°

(2x+

16

)°(7

x)°

(7x)°

(4x)°

(4x)°

(7x)°

(7x)°

1

260

900

1440

5

6

1

2

m

∠A

= 1

15,

m∠

B =

65,

m

∠L =

100,

m∠

M =

110,

m∠

C =

115,

m∠

D =

65

m

∠N

= 7

0,

m∠

P =

80

m

∠S

= 1

16,

m∠

T =

116,

m∠

D =

140,

m∠

E =

140,

m

∠W

= 6

4,

m∠

U =

64

m∠

I =

80,

m∠

F =

80,

m∠

H =

140,

m∠

G =

140

9

0,

90

108,

72

150,

30

4

5

40

30

Answers (Lesson 6-1)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

8

Gle

nc

oe

Ge

om

etr

y

Fin

d t

he

su

m o

f th

e m

ea

su

re

s o

f th

e i

nte

rio

r a

ng

les o

f e

ach

co

nv

ex

po

lyg

on

.

1

. 11-g

on

2. 14-g

on

3. 17-g

on

Th

e m

ea

su

re

of

an

in

terio

r a

ng

le o

f a

re

gu

lar p

oly

go

n i

s g

ive

n.

Fin

d t

he

nu

mb

er o

f sid

es i

n t

he

po

lyg

on

.

4. 144

5. 156

6. 160

Fin

d t

he

me

asu

re

of

ea

ch

in

terio

r a

ng

le.

7

.

8.

Fin

d t

he

me

asu

re

s o

f a

n e

xte

rio

r a

ng

le a

nd

an

in

terio

r a

ng

le g

ive

n t

he

nu

mb

er o

f

sid

es o

f e

ach

re

gu

lar p

oly

go

n.

Ro

un

d t

o t

he

ne

are

st

ten

th,

if n

ece

ssa

ry

.

9

. 16

10

. 24

11

. 30

12

. 14

13

. 22

14

. 40

15

. C

RY

ST

ALLO

GR

AP

HY

C

ryst

als

are

cla

ssif

ied

acc

ord

ing t

o s

even

cry

stal

syst

em

s. T

he

basi

s of

the c

lass

ific

ati

on

is

the s

hap

es

of

the f

ace

s of

the c

ryst

al.

Tu

rqu

ois

e b

elo

ngs

to

the t

ricl

inic

syst

em

. E

ach

of

the s

ix f

ace

s of

turq

uois

e i

s in

th

e s

hap

e o

f a p

ara

llelo

gra

m.

Fin

d t

he s

um

of

the m

easu

res

of

the i

nte

rior

an

gle

s of

on

e s

uch

face

.

Practi

ce

An

gle

s o

f P

oly

go

ns

6 -1

x°

( 2x+

15

) °( 3

x-

20

) °

( x+

15

) °

JK

MN

(6x-

4)°

(2x+

8)°

(6x-

4)°

(2x+

8)°

m

∠R

= 1

28,

m∠

S =

52

m∠

T =

128,

m∠

U =

52

1

620

2160

2700

1

0

15

18

m

∠J =

115,

m∠

K =

130,

m∠

M =

50,

m∠

N =

65

1

57.5

, 22.5

1

65,

15

168,

12

1

54.3

, 25.7

1

63.6

, 16.4

1

71,

9

3

60

Lesson 6-1


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

9

Gle

nc

oe

Ge

om

etr

y

Wo

rd

Pro

ble

m P

racti

ce

An

gle

s o

f P

oly

go

ns

La

Trib

un

a

1

. A

RC

HIT

EC

TU

RE

In

th

e U

ffiz

i gall

ery

in

Flo

ren

ce,

Italy

, th

ere

is

a r

oom

bu

ilt

by B

uon

tale

nti

call

ed

th

e T

ribu

ne (

La

Trib

un

a i

n I

tali

an

). T

his

room

is

shap

ed

li

ke a

regu

lar

oct

agon

.

Wh

at

an

gle

do c

on

secu

tive w

all

s of

the

Tri

bu

ne m

ak

e w

ith

each

oth

er?

2

. B

OX

ES

Jasm

ine i

s d

esi

gn

ing b

oxes

she

wil

l u

se t

o s

hip

her

jew

elr

y.

Sh

e w

an

ts

to s

hap

e t

he b

ox l

ike a

regu

lar

poly

gon

. In

ord

er

for

the b

oxes

to p

ack

tig

htl

y,

she d

eci

des

to u

se a

regu

lar

poly

gon

th

at

has

the p

rop

ert

y t

hat

the m

easu

re o

f it

s in

teri

or

an

gle

s is

half

th

e m

easu

re o

f it

s exte

rior

an

gle

s. W

hat

regu

lar

poly

gon

sh

ou

ld s

he u

se?

3

. T

HE

AT

ER

A

th

eate

r fl

oor

pla

n i

s sh

ow

n

in t

he f

igu

re.

Th

e u

pp

er

five s

ides

are

p

art

of

a r

egu

lar

dod

eca

gon

.

Fin

d m∠

1.

4. A

RC

HE

OLO

GY

A

rch

eolo

gis

ts u

neart

hed

p

art

s of

two a

dja

cen

t w

all

s of

an

an

cien

t ca

stle

.

Befo

re i

t w

as

un

eart

hed

, th

ey k

new

fro

m

an

cien

t te

xts

th

at

the c

ast

le w

as

shap

ed

li

ke a

regu

lar

poly

gon

, bu

t n

obod

y k

new

h

ow

man

y s

ides

it h

ad

. S

om

e s

aid

6,

oth

ers

8,

an

d s

om

e e

ven

said

100.

Fro

m

the i

nfo

rmati

on

in

th

e f

igu

re,

how

man

y

sid

es

did

th

e c

ast

le r

eall

y h

ave?

5

. P

OLY

GO

N P

AT

HIn

Ms.

Ric

kets

’ m

ath

cl

ass

, st

ud

en

ts m

ad

e a

“p

oly

gon

path

” th

at

con

sist

s of

regu

lar

poly

gon

s of

3,

4,

5,

an

d 6

sid

es

join

ed

togeth

er

as

show

n.

a.

Fin

d m∠

2 a

nd

m∠

5.

b.

Fin

d m∠

3 a

nd

m∠

4.

c.

Wh

at

is m∠

1?

6 -1

21

3

4

5

stag

e

1

24˚

1

35°

a

n e

qu

ilate

ral

tria

ng

le

1

20

1

5

90 a

nd

60

162 a

nd

132

96


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

10

G

len

co

e G

eo

me

try

En

rich

men

t

Cen

tral

An

gle

s o

f R

eg

ula

r P

oly

go

ns

You

have l

earn

ed

abou

t th

e i

nte

rior

an

d e

xte

rior

an

gle

s of

a p

oly

gon

. R

egu

lar

poly

gon

s als

o h

ave c

en

tra

l a

ng

les.

A c

en

tral

an

gle

is

measu

red

fro

m t

he c

en

ter

of

the p

oly

gon

.

Th

e c

en

ter

of

a p

oly

gon

is

the p

oin

t equ

idis

tan

t fr

om

all

of

th

e v

ert

ices

of

the p

oly

gon

, ju

st a

s th

e c

en

ter

of

a c

ircl

e i

s th

e p

oin

t equ

idis

tan

t fr

om

all

of

the p

oin

ts o

n t

he c

ircl

e.

Th

e

cen

tral

an

gle

is

the a

ngle

dra

wn

wit

h t

he v

ert

ex a

t th

e

cen

ter

of

the c

ircl

e a

nd

th

e s

ides

of

an

gle

dra

wn

th

rou

gh

co

nse

cuti

ve v

ert

ices

of

the p

oly

gon

. O

ne o

f th

e c

en

tral

an

gle

s th

at

can

be d

raw

n i

n t

his

regu

lar

hexagon

is

∠A

PB

. Y

ou

may r

em

em

ber

from

mak

ing c

ircl

e g

rap

hs

that

there

are

360°

aro

un

d t

he c

en

ter

of

a c

ircl

e.

1. B

y u

sin

g l

ogic

or

by d

raw

ing s

ketc

hes,

fin

d t

he m

easu

re o

f th

e c

en

tral

an

gle

of

each

re

gu

lar

poly

gon

.

2. M

ak

e a

con

ject

ure

abou

t h

ow

th

e m

easu

re o

f a c

en

tral

an

gle

of

a r

egu

lar

poly

gon

rela

tes

to t

he m

easu

res

of

the i

nte

rior

an

gle

s an

d e

xte

rior

an

gle

s of

a r

egu

lar

poly

gon

.

3

. C

HA

LLE

NG

E In

obtu

se △

AB

C,

−−

−

BC

is

the l

on

gest

sid

e.

−−

AC

is

als

o a

sid

e o

f a

21-s

ided

regu

lar

poly

gon

. −

−

AB

is

als

o a

sid

e o

f a 2

8-s

ided

regu

lar

poly

gon

. T

he

21-s

ided

regu

lar

poly

gon

an

d t

he 2

8-s

ided

regu

lar

poly

gon

have t

he s

am

e

cen

ter

poin

t P

. F

ind

n i

f −

−−

BC

is

a s

ide o

f a n

-sid

ed

regu

lar

poly

gon

th

at

has

cen

ter

poin

t P

.

(H

int:

Sk

etc

h a

cir

cle w

ith

cen

ter

P a

nd

pla

ce p

oin

ts A

, B

, an

d C

on

th

e c

ircl

e.)

6 -1

A

B

P

r

1

20

9

0

72

6

0

ab

ou

t 51.4

3

4

5

T

he m

easu

re o

f th

e e

xte

rio

r an

gle

eq

uals

th

e m

easu

re o

f th

e c

en

tral

an

gle

. T

he c

en

tral

an

gle

is s

up

ple

men

tary

to

in

teri

or

an

gle

.

n

= 1

2

Lesson 6-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

11

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Para

llelo

gra

ms

Sid

es

an

d A

ng

les

of

Pa

rall

elo

gra

ms

A q

uad

rila

tera

l w

ith

both

pair

s of

op

posi

te s

ides

para

llel

is a

pa

ra

lle

log

ra

m.

Here

are

fou

r im

port

an

t p

rop

ert

ies

of

para

llelo

gra

ms.

If

ABCD

is a

pa

ra

lle

log

ra

m,

fin

d t

he

va

lue

of

ea

ch

va

ria

ble

.−

−

AB

an

d −

−−

CD

are

op

posi

te s

ides,

so −

−

AB

$ −

−−

CD

.

2a

= 3

4

a

= 1

7

∠A

an

d ∠

C a

re o

pp

osi

te a

ngle

s, s

o ∠

A $

∠C

.

8b =

112

b =

14

Exerc

ises

Fin

d t

he

va

lue

of

ea

ch

va

ria

ble

.

1

.

2.

3

.

4.

5

.

6.

PQ

RS

BA

DC

34

2a

8b°

112°

4y°

3x°

8y

88

6x°

72

x

30

x150

2y

6 -2

If P

QR

S i

s a

pa

rall

elo

gra

m,

the

n

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n its

op

po

site

sid

es a

re c

on

gru

en

t.

−−

−

PQ

$ −

−

SR

an

d −

−

PS

$ −

−−

QR

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n its

op

po

site

an

gle

s a

re c

on

gru

en

t.

∠P

$ ∠

R a

nd

∠S

$ ∠

Q

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n its

co

nse

cu

tive

an

gle

s a

re

su

pp

lem

en

tary

.

∠P

an

d ∠

S a

re s

up

ple

me

nta

ry;

∠S

an

d ∠

R a

re s

up

ple

me

nta

ry;

∠R

an

d ∠

Q a

re s

up

ple

me

nta

ry;

∠Q

an

d ∠

P a

re s

up

ple

me

nta

ry.

If a

pa

ralle

log

ram

ha

s o

ne

rig

ht

an

gle

,

the

n it

ha

s f

ou

r rig

ht

an

gle

s.

If m

∠P

= 9

0,

the

n m

∠Q

= 9

0,

m∠

R =

90

, a

nd

m∠

S =

90

.

Exam

ple

3y

12

6x 60°55°

5x°

2y°

6x°

12

x°

3y°

x

= 3

0;

y =

22.5

x =

15;

y =

11

x

= 2

; y =

4

x =

10;

y =

40

x

= 1

3;

y =

32.5

x =

5;

y =

180

Answers (Lesson 6-1 and Lesson 6-2)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

12

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Para

llelo

gra

ms

Dia

go

na

ls o

f P

ara

lle

log

ram

s T

wo i

mp

ort

an

t p

rop

ert

ies

of

p

ara

llelo

gra

ms

deal

wit

h t

heir

dia

gon

als

.

F

ind

th

e v

alu

e o

f x a

nd

y i

n p

ara

lle

log

ra

m ABCD

.

Th

e d

iagon

als

bis

ect

each

oth

er,

so A

E =

CE

an

d D

E =

BE

.

6x =

24

4y =

18

x =

4

y =

4.5

Exerc

ises

Fin

d t

he

va

lue

of

ea

ch

va

ria

ble

.

1

.

2.

3.

4

.

5.

6.

CO

OR

DIN

AT

E G

EO

ME

TR

Y

Fin

d t

he

co

ord

ina

tes o

f th

e i

nte

rse

cti

on

of

the

d

iag

on

als

of

ABCD

wit

h t

he

giv

en

ve

rti

ce

s.

7. A

(3,

6),

B(5

, 8),

C(3

, −

2),

an

d D

(1,

−4)

8. A

(−4,

3),

B(2

, 3),

C(−

1,

−2),

an

d D

(−7,

−2)

9

. P

RO

OF W

rite

a p

ara

gra

ph

pro

of

of

the f

oll

ow

ing.

G

ive

n:

!A

BC

D

P

ro

ve

: △

AE

D #

△B

EC

AB

CD

P

AB

E

CD

6x

18

24

4y

3x

4y

812

4x

2y

28

3x

30

°10

y

AB

CD

E

2x°6

0°

4y°

3x°

2y

12

6 -2

If A

BC

D i

s a

pa

rall

elo

gra

m,

the

n

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n

its d

iag

on

als

bis

ect

ea

ch

oth

er.

AP

= P

C a

nd

DP

= P

B

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n e

ach

dia

go

na

l se

pa

rate

s t

he

pa

ralle

log

ram

into

tw

o c

on

gru

en

t tr

ian

gle

s.

△A

CD

# △

CA

B a

nd

△A

DB

# △

CB

D

Exam

ple

x

17

4y

x

= 4

; y =

2

x =

7;

y =

14

x =

15;

y =

7.5

x

= 3

1

−

3 ;

y =

10 √

$

3

x =

15;

y =

6 √

$

2

x =

15;

y =

√ $

$

241

D

iag

on

als

of

a p

ara

lle

log

ram

bis

ec

t e

ac

h o

the

r, s

o −

−

AE

& −

−

CE

an

d −

−

BE

& −

−

DE

. O

pp

os

ite

sid

es

of

a p

ara

lle

log

ram

are

co

ng

rue

nt,

th

ere

fore

−−

AD

& −

−

BC

. B

ec

au

se

co

rre

sp

on

din

g p

art

s o

f th

e t

wo

tri

an

gle

s a

re c

on

gru

en

t, t

he

tr

ian

gle

s a

re c

on

gru

en

t b

y S

SS

.

(

3,

2)

(-

2.5

, 0.5

)

Lesson 6-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

13

G

len

co

e G

eo

me

try

Sk

ills

Pra

ctic

e

Para

llelo

gra

ms

ALG

EB

RA

Fin

d t

he

va

lue

of

ea

ch

va

ria

ble

.

1

.

2.

3

.

4.

5

.

6.

CO

OR

DIN

AT

E G

EO

ME

TR

Y F

ind

th

e c

oo

rd

ina

tes o

f th

e i

nte

rse

cti

on

of

the

dia

go

na

ls

of

HJKL

wit

h t

he

giv

en

ve

rti

ce

s.

7

. H

(1,

1),

J(2

, 3),

K(6

, 3),

L(5

, 1)

8. H

(-1,

4),

J(3

, 3),

K(3

, -

2),

L(-

1,

-1)

9. P

RO

OF

W

rite

a p

ara

gra

ph

pro

of

of

the t

heore

m C

on

secu

tive

an

gle

s in

a p

ara

llel

ogra

m

are

su

pp

lem

enta

ry.

DC

BA

6-2

x-

2y

+1

1926

x°44°

y°

4x- 2

3y-

1

y+

5

x + 10

2b

-1

b+

3

3a

-5

2a

a+

14

6a

+4

10b

+19b

+8

(x+

8)°

(3x)°

(y+

9)°

a

= 5

, b

= 4

x =

136,

y =

44

(

3.5

, 2)

(1

, 1)

G

iven

:

AB

CD

P

rove:

∠A

an

d ∠

B a

re s

up

ple

men

tary

.∠

B a

nd

∠C

are

su

pp

lem

en

tary

.∠

C a

nd

∠D

are

su

pp

lem

en

tary

.∠

D a

nd

∠A

are

su

pp

lem

en

tary

.

P

roo

f: W

e a

re g

ive

n

AB

CD

, s

o w

e k

no

w t

ha

t −

−

AB

||

−−

CD

an

d −

−

BC

||

−−

DA

by

the

de

fin

itio

n o

f a

pa

rall

elo

gra

m.

We

als

o k

no

w t

ha

t if

tw

o p

ara

lle

l li

ne

s a

re

cu

t b

y a

tra

ns

ve

rsa

l, t

he

n c

on

se

cu

tiv

e i

nte

rio

r a

ng

les

are

su

pp

lem

en

tary

. S

o,

∠A

an

d ∠

B,

∠B

an

d ∠

C,

∠C

an

d ∠

D,

an

d ∠

D a

nd

∠A

are

pa

irs

of

su

pp

lem

en

tary

an

gle

s.

x

= 4

3,

y =

12

0

x =

4,

y =

3

x

= 2

1,

y =

25

a =

2,

b =

7


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

14

G

len

co

e G

eo

me

try

Practi

ce

Para

llelo

gra

ms

ALG

EB

RA

Fin

d t

he

va

lue

of e

ach

va

ria

ble

.

1

.

2.

3

.

4.

ALG

EB

RA

U

se

RSTU

to

fin

d e

ach

me

asu

re

or v

alu

e.

5

. m∠RST=

6. m∠STU=

7

. m∠TUR=

8. b=

CO

OR

DIN

AT

E G

EO

ME

TR

Y F

ind

th

e c

oo

rd

ina

tes o

f th

e i

nte

rse

cti

on

of

the

dia

go

na

ls

of PRYZ

wit

h t

he

giv

en

ve

rti

ce

s.

9. P

(2,

5),

R(3

, 3),

Y(-

2, -

3),

Z(-

3, -

1)

10

. P

(2,

3),

R(1

, -

2),

Y(-

5, -

7),

Z(-

4, -

2)

11. P

RO

OF

W

rite

a p

ara

gra

ph

pro

of

of

the f

oll

ow

ing.

G

ive

n: !PRST

an

d !

PQVU

Pro

ve

: ∠V

" ∠S

12

. C

ON

ST

RU

CT

ION

M

r. R

od

riqu

ez u

sed

th

e p

ara

llelo

gra

m a

t th

e r

igh

t to

d

esi

gn

a h

err

ingbon

e p

att

ern

for

a p

avin

g s

ton

e.

He w

ill

use

th

e p

avin

g

ston

e f

or

a s

idew

alk

. If

m∠

1 i

s 130,

fin

d m∠

2, m∠

3,

an

d m∠

4.

TU

SR

B30°

23

4b-

1

25°

PR

S

UVQ

T

12

34

6 -2

(4x)

°

(y+

10)°

(2y-

40)°

4x

5y-

83y+

1

x+6

y+3

15

x-

3

12

b+1

2b

a+2

3a-

4

125

(

0,

1)

(-

1.5

, -

2)

P

roo

f: W

e a

re g

iven

P

RS

T a

nd

P

QV

U.

Sin

ce o

pp

osit

e a

ng

les o

f a

para

llelo

gra

m a

re c

on

gru

en

t, ∠

P #

∠V

an

d ∠

P #

∠S

. S

ince c

on

gru

en

ce

of

an

gle

s i

s t

ran

sit

ive, ∠

V #

∠S

by t

he T

ran

sit

ive P

rop

ert

y o

f

co

ng

ruen

ce.

5

0,

130,

50

a

= 3

, b

= 1

x =

30˚,

y =

50˚

x

= 1

8,

y =

9

x =

2,

y =

4.5

125

6

55

Lesson 6-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

15

G

len

co

e G

eo

me

try

1

. W

ALK

WA

Y A

walk

way i

s m

ad

e b

y

ad

join

ing f

ou

r p

ara

llelo

gra

ms

as

show

n.

Are

th

e e

nd

segm

en

ts a

an

d e

para

llel

to

each

oth

er?

Exp

lain

.

2

. D

IST

AN

CE

F

ou

r fr

ien

ds

live a

t th

e f

ou

r co

rners

of

a b

lock

sh

ap

ed

lik

e a

p

ara

llelo

gra

m.

Gra

cie l

ives

3 m

iles

aw

ay

from

Ken

ny.

How

far

ap

art

do T

ere

sa

an

d T

ravis

liv

e f

rom

each

oth

er?

3. S

OC

CE

R F

ou

r so

ccer

pla

yers

are

loca

ted

at

the c

orn

ers

of

a p

ara

llelo

gra

m.

Tw

o

of

the p

layers

in

op

posi

te c

orn

ers

are

th

e g

oali

es.

In

ord

er

for

goali

e A

to b

e

able

to s

ee t

he t

hre

e o

thers

, sh

e m

ust

be a

ble

to s

ee a

cert

ain

an

gle

x i

n h

er

field

of

vis

ion

.

Wh

at

an

gle

does

the o

ther

goali

e h

ave t

o

be a

ble

to s

ee i

n o

rder

to k

eep

an

eye o

n

the o

ther

thre

e p

layers

?

4

. V

EN

N D

IAG

RA

MS

M

ak

e a

Ven

n

dia

gra

m s

how

ing t

he r

ela

tion

ship

betw

een

squ

are

s, r

ect

an

gle

s, a

nd

p

ara

llelo

gra

ms.

5

. S

KY

SC

RA

PE

RS

O

n v

aca

tion

, T

on

y’s

fa

mil

y t

ook

a h

eli

cop

ter

tou

r of

the c

ity.

Th

e p

ilot

said

th

e

new

est

bu

ild

ing

in t

he c

ity w

as

the b

uil

din

g w

ith

th

is t

op

vie

w.

He

told

Ton

y t

hat

the

exte

rior

an

gle

by

the f

ron

t en

tran

ce

is 7

2°.

Ton

y

wan

ted

to k

now

m

ore

abou

t th

e b

uil

din

g,

so h

e d

rew

th

is

dia

gra

m a

nd

use

d h

is g

eom

etr

y s

kil

ls t

o

learn

a f

ew

more

th

ings.

Th

e f

ron

t en

tran

ce i

s n

ext

to v

ert

ex B

.

a.

Wh

at

are

th

e m

easu

res

of

the f

ou

r an

gle

s of

the p

ara

llelo

gra

m?

b.

How

man

y p

air

s of

con

gru

en

t tr

ian

gle

s are

th

ere

in

th

e f

igu

re?

Wh

at

are

th

ey?

Wo

rd

Pro

ble

m P

racti

ce

Para

llelo

gra

ms

AB

CD

E

72˚

par

alle

logra

ms

rect

ang

les

squ

ares

Goal

ie B

Pla

yer

A

Go

alie

A

x

Pla

yer

B

Tere

saTr

avis

Gra

cie

Ken

ny

a

e

6 -2

Y

es.

Op

po

sit

e s

ides o

f a

para

llelo

gra

m a

re p

ara

llel

an

d t

he

para

llel

pro

pert

y i

s t

ran

sit

ive.

3

mi

H

e o

r sh

e a

lso

has t

o s

ee

an

gle

x.

72,

72,

108,

108

4 p

air

s;

△A

BE

# △

DC

E,

△A

BC

# △

DC

B,

△A

CE

# △

DB

E,

an

d

△A

BD

# △

DC

A


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

16

G

len

co

e G

eo

me

try

En

rich

men

t

Dia

go

nals

of

Para

llelo

gra

ms

In s

om

e d

raw

ings

the d

iagon

al

of

a p

ara

llelo

gra

m a

pp

ears

to b

e t

he a

ngle

bis

ect

or

of

both

op

posi

te a

ngle

s. W

hen

mig

ht

that

be t

rue?

1. G

ive

n:

Para

llelo

gra

m P

QR

S w

ith

dia

gon

al

−−

PR

.

−−

PR

is

an

an

gle

bis

ect

or

of

∠Q

PS

an

d ∠

QR

S.

W

hat

typ

e o

f p

ara

llelo

gra

m i

s P

QR

S?

Ju

stif

y

you

r an

swer.

2. G

ive

n:

Para

llelo

gra

m W

PR

K w

ith

an

gle

bis

ect

or

−−

−

KD

, D

P =

5,

an

d W

D =

7.

F

ind

WK

an

d K

R.

3. R

efe

r to

Exerc

ise 2

. W

rite

a s

tate

men

t abou

t p

ara

llelo

gra

m W

PR

K a

nd

an

gle

bis

ect

or

−−

−

KD

.

4. G

ive

n:

Para

llelo

gra

m A

BC

D w

ith

d

iagon

al

−−

−

BD

an

d a

ngle

bis

ect

or

−−

BP

. P

D =

5,

BP

= 6

, an

d C

P =

6.

Th

e p

eri

mete

r of

tria

ngle

PC

D i

s 15.

F

ind

AB

an

d B

C.

P

QR

S

PD

A

BC

PD

R

K

W

6 -2

∠

QR

S !

∠Q

PS

op

po

sit

e a

ng

les o

f a

para

llelo

gra

m

∠

QP

R !

∠R

PS

defi

nit

ion

of

an

gle

bis

ecto

r

∠

QR

P !

∠S

RP

defi

nit

ion

of

an

gle

bis

ecto

r

∠

QP

R !

∠P

RS

alt

ern

ate

in

teri

or

an

gle

s

∠

RP

S !

∠P

RS

Tra

nsit

ive P

rop

ert

y

−−

SP

! −

−

SR

Is

oscele

s T

rian

gle

Th

eo

rem

S

o a

ll s

ides a

re !

an

d Q

RP

S i

s a

rh

om

bu

s

W

K =

7,

KR

= 1

2

S

am

ple

an

sw

er:

Para

llelo

gra

m

WK

RP

is n

ot

a r

ho

mb

us a

nd

−−

KD

is n

ot

a d

iag

on

al.

A

B =

4,

BC

= 9


NA

ME

DA

TE

PE

RIO

D

Lesson 6-3

Ch

ap

ter

6

17

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Tests

fo

r P

ara

llelo

gra

ms

Co

nd

itio

ns

for

Pa

rall

elo

gra

ms

Th

ere

are

man

y w

ays

to

est

abli

sh t

hat

a q

uad

rila

tera

l is

a p

ara

llelo

gra

m.

F

ind

x a

nd

y s

o t

ha

t FGHJ

is a

pa

ra

lle

log

ra

m.

FG

HJ

is

a p

ara

llelo

gra

m i

f th

e l

en

gth

s of

the o

pp

osi

te

sid

es

are

equ

al.

6x +

3 =

15

4x -

2y =

2

6x =

12

4(2

) -

2y =

2

x =

2

8 -

2y =

2

-2y =

-6

y =

3

Exerc

ises

Fin

d x

an

d y

so

th

at

the

qu

ad

ril

ate

ra

l is

a p

ara

lle

log

ra

m.

1

.

2.

3

.

4.

5

.

6.

AB

CD

E

JHG

F6

x+

3

15

4x

- 2

y2

2x

-2

2y

8

12

11

x°

5y°

55

°

5x

°5

y°

25

°

( x+

y) °

2x

°

30

°

24

°

6y

°

3x

°

9x

°6

y4

5°

18

6 -3

If:

If:

bo

th p

airs o

f o

pp

osite

sid

es a

re p

ara

llel,

−−

AB

||

−−−

DC

an

d −

−

AD

||

−−−

BC

,

bo

th p

airs o

f o

pp

osite

sid

es a

re c

on

gru

en

t, −

−

AB

# −−

− D

C a

nd

−−

AD

# −−

− B

C ,

bo

th p

airs o

f o

pp

osite

an

gle

s a

re c

on

gru

en

t,∠

AB

C #

∠A

DC

an

d ∠

DA

B #

∠B

CD

,

the

dia

go

na

ls b

ise

ct

ea

ch

oth

er,

−−

AE

# −

−

CE

an

d −

−

DE

# −

−

BE

,

on

e p

air o

f o

pp

osite

sid

es is c

on

gru

en

t a

nd

pa

ralle

l, −

−

AB

||

−−−

CD

an

d −

−

AB

# −−

− C

D ,

or

−−

AD

||

−−−

BC

an

d −

−

AD

# −−

− B

C ,

the

n:

the

fig

ure

is a

pa

ralle

log

ram

.th

en

: A

BC

D is a

pa

ralle

log

ram

.

Exam

ple

x =

7;

y =

4x =

5;

y =

25

x =

31;

y =

5x =

5;

y =

3

x =

15;

y =

9x =

30;

y =

15


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

18

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Tests

fo

r P

ara

llelo

gra

ms

Pa

rall

elo

gra

ms

on

th

e C

oo

rdin

ate

Pla

ne

O

n t

he c

oord

inate

pla

ne,

the D

ista

nce

, S

lop

e,

an

d M

idp

oin

t F

orm

ula

s ca

n b

e u

sed

to t

est

if

a q

uad

rila

tera

l is

a p

ara

llelo

gra

m.

D

ete

rm

ine

wh

eth

er A

BC

D i

s a

pa

ra

lle

log

ra

m.

Th

e v

ert

ices

are

A(-

2,

3),

B(3

, 2),

C(2

, -

1),

an

d D

(-3,

0).

Meth

od

1:

Use

th

e S

lop

e F

orm

ula

, m

= y

2 -

y1

−

x2 -

x1

.

slop

e o

f −−

− A

D =

3 -

0

−

-2 -

(-

3)

= 3

−

1 =

3

slop

e o

f −−

− B

C =

2 -

(-

1)

−

3 -

2

=

3

−

1 =

3

slop

e o

f −

−

AB

=

2 -

3

−

3 -

(-

2)

= -

1

−

5

slop

e o

f −−

− C

D =

-

1 -

0

−

2 -

(-

3)

= -

1

−

5

Sin

ce o

pp

osi

te s

ides

have t

he s

am

e s

lop

e,

−−

AB

||

−−−

CD

an

d −−

− A

D |

| −−

− B

C .

Th

ere

fore

, A

BC

D i

s a

para

llelo

gra

m b

y d

efi

nit

ion

.

Meth

od

2:

Use

th

e D

ista

nce

Form

ula

, d

= √

##

##

##

##

#

(x

2 -

x1)2

+ (

y2 -

y1)2

.

AB

= √

##

##

##

##

#

(-

2 -

3)2

+ (

3 -

2)2

= √

##

#

25 +

1 o

r √

##

26

CD

= √

##

##

##

##

##

(2

- (

-3))

2 +

(-

1 -

0)2

= √

##

#

25 +

1 o

r √

##

26

AD

= √

##

##

##

##

##

(-

2 -

(-

3))

2 +

(3 -

0)2

= √

##

#

1 +

9 o

r √

##

10

BC

= √

##

##

##

##

#

(3

- 2

)2 +

(2 -

(-

1))

2 =

√ #

##

1 +

9 o

r √

##

10

Sin

ce b

oth

pair

s of

op

posi

te s

ides

have t

he s

am

e l

en

gth

, −

−

AB

$ −−

− C

D a

nd

−−−

AD

$ −−

− B

C .

Th

ere

fore

, A

BC

D i

s a p

ara

llelo

gra

m b

y T

heore

m 6

.9.

Exerc

ises

Gra

ph

ea

ch

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s.

De

term

ine

wh

eth

er t

he

fig

ure

is

a p

ara

lle

log

ra

m.

Ju

sti

fy y

ou

r a

nsw

er w

ith

th

e m

eth

od

in

dic

ate

d.

1

. A

(0,

0),

B(1

, 3),

C(5

, 3),

D(4

, 0);

2

. D

(-1,

1),

E(2

, 4),

F(6

, 4),

G(3

, 1);

Slo

pe F

orm

ula

S

lop

e F

orm

ula

3. R

(-1,

0),

S(3

, 0),

T(2

, -

3),

U(-

3,

-2);

4. A

(-3,

2),

B(-

1,

4),

C(2

, 1),

D(0

, -

1);

Dis

tan

ce F

orm

ula

D

ista

nce

an

d S

lop

e F

orm

ula

s

5

. S

(-2,

4),

T(-

1,

-1),

U(3

, -

4),

V(2

, 1);

6

. F

(3,

3),

G(1

, 2),

H(-

3,

1),

I(-

1,

4);

Dis

tan

ce a

nd

Slo

pe F

orm

ula

s M

idp

oin

t F

orm

ula

7. A

para

llelo

gra

m h

as

vert

ices

R(-

2,

-1),

S(2

, 1),

an

d T

(0,

-3).

Fin

d a

ll p

oss

ible

co

ord

inate

s fo

r th

e f

ou

rth

vert

ex.

x

y

O

C

A

D

B

6 -3 Exam

ple

y

es

yes

n

o

yes

y

es

n

o

(

4,

-1),

(0,

3),

or

(-4,

-5)

S

ee s

tud

en

ts’

wo

rk


NA

ME

DA

TE

PE

RIO

D

Lesson 6-3

Ch

ap

ter

6

19

G

len

co

e G

eo

me

try

Sk

ills

Pra

ctic

e

Tests

fo

r P

ara

llelo

gra

ms

De

term

ine

wh

eth

er e

ach

qu

ad

ril

ate

ra

l is

a p

ara

lle

log

ra

m.

Ju

sti

fy y

ou

r a

nsw

er.

1.

2.

3.

4.

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s.

De

term

ine

wh

eth

er t

he

fig

ure

is a

pa

ra

lle

log

ra

m.

Ju

sti

fy y

ou

r a

nsw

er w

ith

th

e

me

tho

d i

nd

ica

ted

.

5. P

(0,

0),

Q(3

, 4),

S(7

, 4),

Y(4

, 0);

Slo

pe F

orm

ula

6. S

(-2,

1),

R(1

, 3),

T(2

, 0),

Z(-

1,

-2);

Dis

tan

ce a

nd

Slo

pe F

orm

ula

s

7. W

(2,

5),

R(3

, 3),

Y(-

2,

-3),

N(-

3,

1);

Mid

poin

t F

orm

ula

ALG

EB

RA

F

ind

x a

nd

y s

o t

ha

t e

ach

qu

ad

ril

ate

ra

l is

a p

ara

lle

log

ra

m.

8.

9.

10

.

11

.

x+

16

y+

19

2y

2x

-8

2y-

11

y+

3

4x-

3

3x

( 4x

- 3

5) °

( 3x

+ 1

0) °

( 2y

- 5

) °

( y+

15) °

3x

- 1

4

x+

20

3y

+ 2

y+

20

6 -3

Y

es;

a p

air

of

op

po

sit

e s

ides

Yes;

bo

th p

air

s o

f o

pp

osit

e

is p

ara

llel

an

d c

on

gru

en

t.

an

gle

s a

re c

on

gru

en

t.

N

o;

no

ne o

f th

e t

ests

fo

r

Y

es;

bo

th p

air

s o

f o

pp

osit

e s

ides

para

llelo

gra

ms i

s f

ulf

ille

d.

are

co

ng

ruen

t.

Y

es;

SR

= Z

T a

nd

th

e s

lop

es o

f −

−

SR

an

d −

−

ZT a

re e

qu

al, s

o o

ne p

air

of

op

po

sit

e s

ides i

s p

ara

llel

an

d c

og

ruen

t.

x

= 2

4,

y =

19

x =

3,

y =

14

x

= 4

5,

y =

20

x =

17,

y =

9

N

o;

the m

idp

oin

ts o

f th

e d

iag

on

als

are

no

t th

e s

am

e p

oin

t.

Y

es;

the s

lop

e o

f −

−

PY

an

d −

−

QS

are

eq

ual

an

d t

he s

lop

e o

f −

−

PQ

an

d −

−

YS

are

eq

ual, s

o −

−

PY

‖ −

−

QS

an

d −

−

PQ

‖ −

−

YS

. O

pp

osit

e s

ides a

re p

ara

llel.

S

ee s

tud

en

ts’

gra

ph

s.


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

20

G

len

co

e G

eo

me

try

De

term

ine

wh

eth

er e

ach

qu

ad

ril

ate

ra

l is

a p

ara

lle

log

ra

m.

Ju

sti

fy y

ou

r a

nsw

er.

1.

2.

3

.

4.

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s.

De

term

ine

wh

eth

er t

he

fig

ure

is a

pa

ra

lle

log

ra

m.

Ju

sti

fy y

ou

r a

nsw

er w

ith

th

e

me

tho

d i

nd

ica

ted

.

5. P

(-5,

1),

S(-

2,

2),

F(-

1, -

3),

T(2

, -

2);

Slo

pe F

orm

ula

6

. R

(-2,

5),

O(1

, 3),

M(-

3, -

4),

Y(-

6, -

2);

Dis

tan

ce a

nd

Slo

pe F

orm

ula

s

ALG

EB

RA

F

ind

x a

nd

y s

o t

ha

t th

e q

ua

dril

ate

ra

l is

a p

ara

lle

log

ra

m.

7.

8.

9

.

10.

11

. T

ILE

DE

SIG

N T

he p

att

ern

sh

ow

n i

n t

he f

igu

re i

s to

con

sist

of

con

gru

en

t p

ara

llelo

gra

ms.

How

can

th

e d

esi

gn

er

be c

ert

ain

th

at

the s

hap

es

are

p

ara

llelo

gra

ms?

11

8°

62

°

11

8°

62

°

( 5x

+ 2

9) °

( 7x

- 1

1) °

( 3y

+ 1

5) °

( 5y

- 9

) °

3y-

5

-3x

+ 4

-4x

- 2

2y+

8

-4

x+

6

-6

x

7y

+ 3

12

y-

7-

4y-

2

x+

12

-2x

+ 6

y+

23

Practi

ce

Tests

fo

r P

ara

llelo

gra

ms

6 -3

Y

es;

the d

iag

on

als

bis

ect

No

; n

on

e o

f th

e t

ests

fo

r

each

oth

er.

para

llelo

gra

ms i

s f

ulf

ille

d.

Y

es;

bo

th p

air

s o

f o

pp

osit

e

No

; n

on

e o

f th

e t

ests

fo

r

an

gle

s a

re c

on

gru

en

t.

p

ara

llelo

gra

ms i

s f

ulf

ille

d.

y

es

x

= 2

0,

y =

12

x =

-6,

y =

13

x

= -

3,

y =

2

x =

-2,

y =

-5

S

am

ple

an

sw

er:

Co

nfi

rm t

hat

bo

th p

air

s o

f o

pp

osit

e !

are

#.

y

es

S

ee s

tud

en

ts’

wo

rk

Lesson 6-3


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

21

G

len

co

e G

eo

me

try

1

. B

ALA

NC

ING

N

ikia

, M

ad

ison

, A

ngela

, an

d S

helb

y a

re b

ala

nci

ng t

hem

selv

es

on

an

“X

”-sh

ap

ed

flo

ati

ng o

bje

ct.

To

bala

nce

th

em

selv

es,

th

ey w

an

t to

m

ak

e t

hem

selv

es

the v

ert

ices

of

a

para

llelo

gra

m.

In o

rder

to a

chie

ve t

his

, d

o a

ll f

ou

r of

them

have t

o b

e t

he s

am

e d

ista

nce

fro

m

the c

en

ter

of

the o

bje

ct?

Exp

lain

.

2

. C

OM

PA

SS

ES

T

wo c

om

pass

need

les

pla

ced

sid

e b

y s

ide o

n a

table

are

both

2 i

nch

es

lon

g a

nd

poin

t d

ue n

ort

h.

Do

they f

orm

th

e s

ides

of

a p

ara

llelo

gra

m?

3

. FO

RM

AT

ION

F

ou

r je

ts a

re f

lyin

g i

n

form

ati

on

. T

hre

e o

f th

e j

ets

are

sh

ow

n

in t

he g

rap

h.

If t

he f

ou

r je

ts a

re l

oca

ted

at

the v

ert

ices

of

a p

ara

llelo

gra

m,

wh

at

are

th

e t

hre

e p

oss

ible

loca

tion

s of

the

mis

sin

g j

et?

4

. S

TR

EE

T L

AM

PS

W

hen

a c

oord

inate

p

lan

e i

s p

lace

d o

ver

the H

arr

isvil

le t

ow

n

map

, th

e f

ou

r st

reet

lam

ps

in t

he c

en

ter

are

loca

ted

as

show

n.

Do t

he f

ou

r la

mp

s fo

rm t

he v

ert

ices

of

a p

ara

llelo

gra

m?

Exp

lain

.

5

. P

ICT

UR

E F

RA

ME

Aaro

n i

s m

ak

ing a

w

ood

en

pic

ture

fra

me i

n t

he s

hap

e o

f a

para

llelo

gra

m.

He h

as

two p

iece

s of

wood

th

at

are

3 f

eet

lon

g a

nd

tw

o t

hat

are

4 f

eet

lon

g.

a.

If h

e c

on

nect

s th

e p

iece

s of

wood

at

their

en

ds

to e

ach

oth

er,

in

wh

at

ord

er

mu

st h

e c

on

nect

th

em

to m

ak

e

a p

ara

llelo

gra

m?

b.

How

man

y d

iffe

ren

t p

ara

llelo

gra

ms

cou

ld h

e m

ak

e w

ith

th

ese

fou

r le

ngth

s of

wood

?

c.

Exp

lain

som

eth

ing A

aro

n m

igh

t d

o

to s

peci

fy p

reci

sely

th

e s

hap

e o

f th

e

para

llelo

gra

m.

5

5y

x

O–

5

–5

5

5

y

xO

Shel

by

Mad

iso

n

Nik

ia

Angel

a

6 -3

Wo

rd

Pro

ble

m P

racti

ce

Tests

fo

r P

ara

llelo

gra

ms

N

o,

Mad

iso

n a

nd

An

gela

have t

o

be t

he s

am

e d

ista

nce a

nd

Nik

ia

an

d S

helb

y h

ave t

o b

e t

he s

am

e

dis

tan

ce b

ut

Nik

ia a

nd

Sh

elb

y’s

dis

tan

ce f

rom

th

e c

en

ter

do

es

no

t h

ave t

o b

e e

qu

al

to M

ad

iso

n

an

d A

ng

ela

’s d

ista

nce.

y

es

(

1,

7),

(9,

-1),

(-

7,

-3)

Y

es;

the l

en

gth

s o

f th

e o

pp

osit

e

sid

es a

re c

on

gru

en

t.

He m

ust

alt

ern

ate

th

e l

en

gth

s

3,

4,

3,

4 o

r 4,

3,

4,

3.

infi

nit

ely

man

y

Sam

ple

an

sw

er:

He c

ou

ld

sp

ecif

y t

he l

en

gth

of

a

dia

go

nal.


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

22

G

len

co

e G

eo

me

try

En

rich

men

t

Tests

fo

r P

ara

llelo

gra

ms

By d

efi

nit

ion

, a q

uad

rila

tera

l is

a p

ara

llelo

gra

m i

f a

nd

on

ly i

f both

pair

s of

op

posi

te s

ides

are

para

llel.

Wh

at

con

dit

ion

s oth

er

than

both

pair

s of

op

posi

te s

ides

para

llel

wil

l gu

ara

nte

e

that

a q

uad

rila

tera

l is

a p

ara

llelo

gra

m?

In t

his

act

ivit

y,

severa

l p

oss

ibil

itie

s w

ill

be

invest

igate

d b

y d

raw

ing q

uad

rila

tera

ls t

o s

ati

sfy c

ert

ain

con

dit

ion

s. R

em

em

ber

that

an

y

test

th

at

seem

s to

work

is

not

gu

ara

nte

ed

to w

ork

un

less

it

can

be f

orm

all

y p

roven

.

Co

mp

lete

.

1. D

raw

a q

uad

rila

tera

l w

ith

on

e p

air

of

op

posi

te s

ides

con

gru

en

t.

Mu

st i

t be a

para

llelo

gra

m?

2. D

raw

a q

uad

rila

tera

l w

ith

both

pair

s of

op

posi

te s

ides

con

gru

en

t.

Mu

st i

t be a

para

llelo

gra

m?

3. D

raw

a q

uad

rila

tera

l w

ith

on

e p

air

of

op

posi

te s

ides

para

llel

an

d

the o

ther

pair

of

op

posi

te s

ides

con

gru

en

t. M

ust

it

be a

p

ara

llelo

gra

m?

4. D

raw

a q

uad

rila

tera

l w

ith

on

e p

air

of

op

posi

te s

ides

both

para

llel

an

d c

on

gru

en

t. M

ust

it

be a

para

llelo

gra

m?

5. D

raw

a q

uad

rila

tera

l w

ith

on

e p

air

of

op

posi

te a

ngle

s co

ngru

en

t.

Mu

st i

t be a

para

llelo

gra

m?

6. D

raw

a q

uad

rila

tera

l w

ith

both

pair

s of

op

posi

te a

ngle

s co

ngru

en

t.

Mu

st i

t be a

para

llelo

gra

m?

7. D

raw

a q

uad

rila

tera

l w

ith

on

e p

air

of

op

posi

te s

ides

para

llel

an

d

on

e p

air

of

op

posi

te a

ngle

s co

ngru

en

t. M

ust

it

be a

para

llelo

gra

m?

6 -3

no

yes

no

yes

no

yes

yes

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

23

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Recta

ng

les

Pro

pe

rtie

s o

f R

ect

an

gle

s A

re

cta

ng

le i

s a q

uad

rila

tera

l w

ith

fou

r

righ

t an

gle

s. H

ere

are

th

e p

rop

ert

ies

of

rect

an

gle

s.

A r

ect

an

gle

has

all

th

e p

rop

ert

ies

of

a p

ara

llelo

gra

m.

• O

pp

osi

te s

ides

are

para

llel.

• O

pp

osi

te a

ngle

s are

con

gru

en

t.

• O

pp

osi

te s

ides

are

con

gru

en

t.

• C

on

secu

tive a

ngle

s are

su

pp

lem

en

tary

.

• T

he d

iagon

als

bis

ect

each

oth

er.

Als

o:

• A

ll f

ou

r an

gle

s are

rig

ht

an

gle

s.

∠U

TS

, ∠

TS

R,

∠S

RU

, an

d ∠

RU

T a

re r

igh

t an

gle

s.

• T

he d

iagon

als

are

con

gru

en

t.

−−

TR

# −

−−

US

TS R

U

Q

Q

ua

dril

ate

ra

l R

UT

S

ab

ov

e i

s a

re

cta

ng

le.

If U

S =

6x +

3 a

nd

RT

= 7

x -

2,

fin

d x

.

Th

e d

iagon

als

of

a r

ect

an

gle

are

con

gru

en

t,

so U

S =

RT

.

6x +

3 =

7x -

2

3 =

x -

2

5 =

x

Q

ua

dril

ate

ra

l R

UT

S

ab

ov

e i

s a

re

cta

ng

le.

If m∠

ST

R =

8x +

3

an

d m∠

UT

R =

16x

- 9

, fi

nd

m∠

ST

R.

∠U

TS

is

a r

igh

t an

gle

, so

m

∠S

TR

+ m

∠U

TR

= 9

0.

8x +

3 +

16

x -

9 =

90

24

x -

6 =

90

24

x =

96

x =

4

m∠

ST

R =

8x +

3 =

8(4

) +

3 o

r 35

Exerc

ises

Qu

ad

ril

ate

ra

l A

BC

D i

s a

re

cta

ng

le.

1. If

AE

= 3

6 a

nd

CE

= 2

x -

4,

fin

d x

.

2. If

BE

= 6

y +

2 a

nd

CE

= 4

y +

6,

fin

d y

.

3. If

BC

= 2

4 a

nd

AD

= 5

y -

1,

fin

d y

.

4. If

m∠

BE

A =

62,

fin

d m

∠B

AC

.

5. If

m∠

AE

D =

12

x a

nd

m∠

BE

C =

10

x +

20,

fin

d m

∠A

ED

.

6. If

BD

= 8

y -

4 a

nd

AC

= 7

y +

3,

fin

d B

D.

7. If

m∠

DB

C =

10

x a

nd

m∠

AC

B =

4x

2-

6,

fin

d m

∠ A

CB

.

8. If

AB

= 6

y a

nd

BC

= 8

y,

fin

d B

D i

n t

erm

s of

y.

BC D

A

E

6 -4

Exam

ple

1Exam

ple

2

20

2

5

59

120

52

30

10

y


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

24

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Recta

ng

les

Pro

ve

th

at

Pa

rall

elo

gra

ms

Are

Re

cta

ng

les

Th

e d

iagon

als

of

a r

ect

an

gle

are

co

ngru

en

t, a

nd

th

e c

on

vers

e i

s als

o t

rue.

If t

he

dia

go

na

ls o

f a

pa

ralle

log

ram

are

co

ng

rue

nt,

th

en

th

e p

ara

llelo

gra

m is a

re

cta

ng

le.

In t

he c

oord

inate

pla

ne y

ou

can

use

th

e D

ista

nce

Form

ula

, th

e S

lop

e F

orm

ula

, an

d

pro

pert

ies

of

dia

gon

als

to s

how

th

at

a f

igu

re i

s a r

ect

an

gle

.

Q

ua

dril

ate

ra

l A

BC

D h

as v

erti

ce

s A

(-3

, 0

), B

(-2

, 3

),

C(4

, 1

), a

nd

D(3

, -

2).

De

term

ine

wh

eth

er A

BC

D i

s a

re

cta

ng

le.

Meth

od

1:

Use

th

e S

lop

e F

orm

ula

.

slop

e o

f −

−

AB

=

3 -

0

−

-2 -

(-

3)

= 3

−

1 o

r 3

slop

e o

f −−

− A

D =

-

2 -

0

−

3 -

(-

3)

= -

2

−

6 o

r -

1

−

3

slop

e o

f −−

− C

D =

-

2 -

1

−

3 -

4

= -

3

−

-1 o

r 3

slop

e o

f −−

− B

C =

1 -

3

−

4 -

(-

2)

= -

2

−

6 o

r -

1

−

3

Op

posi

te s

ides

are

para

llel,

so t

he f

igu

re i

s a p

ara

llelo

gra

m.

Con

secu

tive s

ides

are

p

erp

en

dic

ula

r, s

o A

BC

D i

s a r

ect

an

gle

.

x

y O

D

B

A

EC

6 -4 Exam

ple

Me

tho

d 2

: U

se t

he D

ista

nce

Form

ula

.

AB

= √

##

##

##

##

##

(-

3 -

(-

2) )

2 +

(0 -

3 ) 2

or

√ #

#

10

BC

= √

##

##

##

##

(-

2 -

4 ) 2

+ (

3 -

1 ) 2

or

√ #

#

40

CD

= √

##

##

##

##

#

(4

- 3

) 2 +

(1 -

(-

2) )

2

or

√ #

#

10

AD

= √

##

##

##

##

##

(-3 -

3 ) 2

+ (

0 -

(-

2) )

2

or

√ #

#

40

Op

posi

te s

ides

are

con

gru

en

t, t

hu

s A

BC

D i

s a p

ara

llelo

gra

m.

AC

= √

##

##

##

##

(-

3 -

4 ) 2

+ (

0 -

1 ) 2

or

√ #

#

50

BD

= √

##

##

##

##

##

(-2 -

3 ) 2

+ (

3 -

(-

2) )

2

or

√ #

#

50

AB

CD

is

a p

ara

llelo

gra

m w

ith

con

gru

en

t d

iagon

als

, so

AB

CD

is

a r

ect

an

gle

.

Exerc

ises

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s.

De

term

ine

wh

eth

er t

he

fig

ure

is a

re

cta

ng

le.

Ju

sti

fy y

ou

r a

nsw

er u

sin

g t

he

ind

ica

ted

fo

rm

ula

.

1

. A

(−3,

1),

B(−

3,

3),

C(3

, 3),

D(3

, 1);

Dis

tan

ce F

orm

ula

2

. A

(−3,

0),

B(−

2,

3),

C(4

, 5),

D(3

, 2);

Slo

pe F

orm

ula

3

. A

(−3,

0),

B(−

2,

2),

C(3

, 0),

D(2

−2);

Dis

tan

ce F

orm

ula

4. A

(−1,

0),

B(0

, 2),

C(4

, 0),

D(3

, −

2);

Dis

tan

ce F

orm

ula

Y

es;

AB

= 2

, B

C =

6,

CD

= 2

, D

A =

6,

AC

= √

""

40 ,

BD

= √

""

40 ,

op

po

sit

e

sid

es a

nd

dia

go

nals

are

co

ng

ruen

t.

N

o;

slo

pe o

f −

−

AB

= 3

, slo

pe o

f −

−

BC

= 1

−

3 ,

slo

pes s

ho

w t

hat

two

co

nsecu

tive

sid

es a

re n

ot

perp

en

dic

ula

r.

S

ee s

tud

en

ts’

wo

rk

N

o;

AC

= 6

, B

D =

√ "

"

32 ,

dia

go

nals

are

no

t co

ng

ruen

t.

CD

= √

"

5 ,

DA

= √

""

20 ,

AC

= 5

, B

D =

5,

op

po

sit

e s

ides a

nd

dia

go

nals

are

co

ng

ruen

t.

Y

es;

AB

= √

"

5 ,

BC

= √

""

20 ,

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

25

G

len

co

e G

eo

me

try

Sk

ills

Pra

ctic

e

Recta

ng

les

ALG

EB

RA

Q

ua

dril

ate

ra

l A

BC

D i

s a

re

cta

ng

le.

1. If

AC

= 2

x +

13 a

nd

DB

= 4

x -

1,

fin

d D

B.

2. If

AC

= x

+ 3

an

d D

B =

3x -

19,

fin

d A

C.

3. If

AE

= 3

x +

3 a

nd

EC

= 5

x -

15,

fin

d A

C.

4. If

DE

= 6

x -

7 a

nd

AE

= 4

x +

9,

fin

d D

B.

5. If

m∠

DA

C =

2x +

4 a

nd

m∠

BA

C =

3x +

1,

fin

d m

∠B

AC

.

6. If

m∠

BD

C =

7x +

1 a

nd

m∠

AD

B =

9x -

7,

fin

d m

∠B

DC

.

7. If

m∠

AB

D =

7x -

31 a

nd

m∠

CD

B =

4x +

5,

fin

d m

∠A

BD

.

8. If

m∠

BA

C =

x +

3 a

nd

m∠

CA

D =

x +

15,

fin

d m

∠B

AC

.

9

. PR

OO

F:

Wri

te a

tw

o-c

olu

mn

pro

of.

G

ive

n:

RS

TV

is

a r

ect

an

gle

an

d U

is

the

mid

poin

t of

−−

VT

.

P

ro

ve

: △

RU

V '

△S

UT

P

roo

f

S

tate

me

nts

R

ea

so

ns

1

. R

STV

is a

recta

ng

le.

1.

Giv

en

2

. ∠

V a

nd

∠T a

re r

igh

t an

gle

s.

2.

Defi

nit

ion

of

recta

ng

le

3.

∠V

% ∠

T

3.

All r

t &

are

%.

4.

U i

s t

he m

idp

oin

t o

f −

−

VT .

4.

Giv

en

5

. −

−

VU

% −

−

TU

5.

Defi

nit

ion

of

mid

po

int

6.

−−

VR

% −

−

TS

6.

Op

p s

ides o

f '

are

co

ng

ruen

t.

7

. △

RU

V %

△S

UT

7.

SA

S

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s.

De

term

ine

wh

eth

er t

he

fig

ure

is a

re

cta

ng

le.

Ju

sti

fy y

ou

r a

nsw

er u

sin

g t

he

in

dic

ate

d f

orm

ula

.

10. P

(-3,

-2),

Q(-

4,

2),

R(2

, 4),

S(3

, 0);

Slo

pe F

orm

ula

11

. J

(-6,

3),

K(0

, 6),

L(2

, 2),

M(-

4,

-1);

Dis

tan

ce F

orm

ula

12

. T

(4,

1),

U(3

, -

1),

X(-

3,

2),

Y(-

2,

4);

Dis

tan

ce F

orm

ula

DCB

AE

6-4

27

14 60

82

52

43 5

3

39

N

o;

Sam

ple

an

sw

er:

An

gle

s a

re n

ot

rig

ht

an

gle

s.

Y

es;

Sam

ple

an

sw

er:

Bo

th p

air

s o

f o

pp

osit

e s

ides a

re c

on

gru

en

t

an

d d

iag

on

als

are

co

ng

ruen

t.

Y

es;

Sam

ple

an

sw

er:

Bo

th p

air

s o

f o

pp

osit

e s

ides a

re c

on

gru

en

t an

d t

he

dia

go

nals

are

co

ng

ruen

t.

S

ee s

tud

en

ts’

gra

ph

s.


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

26

G

len

co

e G

eo

me

try

Practi

ce

Recta

ng

les

ALG

EB

RA

Q

ua

dril

ate

ra

l RSTU

is a

re

cta

ng

le.

1. If

UZ=x+

21 a

nd

ZS=

3x-

15,

fin

d US

.

2. If

RZ

= 3x +

8 a

nd

ZS

= 6x -

28,

fin

d UZ

.

3. If

RT

= 5x +

8 a

nd

RZ

= 4x +

1,

fin

d ZT

.

4. If

m∠SUT

= 3x +

6 a

nd

m∠RUS

= 5x -

4,

fin

d m∠SUT

.

5. If

m∠SRT

= x

+ 9

an

d m∠UTR

= 2x -

44,

fin

d m∠UTR

.

6. If

m∠RSU

= x

+ 4

1 a

nd

m∠TUS

= 3x +

9,

fin

d m∠RSU

.

Qu

ad

ril

ate

ra

l GHJK

is a

re

cta

ng

le.

Fin

d e

ach

me

asu

re

if m∠

1 =

37

.

7. m∠

2

8. m∠

3

9. m∠

4

10. m∠

5

11

. m∠

6

12. m∠

7

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ra

ph

ea

ch

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s.

De

term

ine

wh

eth

er t

he

fig

ure

is a

re

cta

ng

le.

Ju

sti

fy y

ou

r a

nsw

er u

sin

g t

he

ind

ica

ted

fo

rm

ula

.

13. B

(-4,

3),

G(-

2,

4),

H(1

, -

2),

L(-

1, -

3);

Slo

pe F

orm

ula

14

. N

(-4,

5),

O(6

, 0),

P(3

, -

6),

Q(-

7, -

1);

Dis

tan

ce F

orm

ula

15

. C

(0,

5),

D(4

, 7),

E(5

, 4),

F(1

, 2);

Slo

pe F

orm

ula

16. LA

ND

SC

AP

ING

H

un

tin

gto

n P

ark

off

icia

ls a

pp

roved

a r

ect

an

gu

lar

plo

t of

lan

d f

or

a

Jap

an

ese

Zen

gard

en

. Is

it

suff

icie

nt

to k

now

th

at

op

posi

te s

ides

of

the g

ard

en

plo

t are

co

ngru

en

t an

d p

ara

llel

to d

ete

rmin

e t

hat

the g

ard

en

plo

t is

rect

an

gu

lar?

Exp

lain

.

UTS

RZ

K

21

5

43

67

JHG

6 -4

78

44

9

39

62

57

53

37

37

53

106

74

Y

es;

sam

ple

an

sw

er:

Op

po

sit

e s

ides a

re p

ara

llel

an

d c

on

secu

tive s

ides

are

perp

en

dic

ula

r.

Y

es;

sam

ple

an

sw

er:

Op

po

sit

e s

ides a

re c

on

gru

en

t an

d d

iag

on

als

are

co

ng

ruen

t.

N

o;

sam

ple

an

sw

er:

Dia

go

nals

are

no

t co

ng

ruen

t.

N

o;

if y

ou

on

ly k

no

w t

hat

op

po

sit

e s

ides a

re c

on

gru

en

t an

d p

ara

llel, t

he

mo

st

yo

u c

an

co

nclu

de i

s t

hat

the p

lot

is a

para

llelo

gra

m.

S

ee s

tud

en

ts’

wo

rk

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

27

G

len

co

e G

eo

me

try

1

. FR

AM

ES

Jale

n m

ak

es

the r

ect

an

gu

lar

fram

e s

how

n.

In o

rder

to m

ak

e s

ure

th

at

it i

s a

rect

an

gle

, Jale

n m

easu

res

the d

ista

nce

s BD

an

d AC

. H

ow

sh

ou

ld t

hese

tw

o

dis

tan

ces

com

pare

if

the f

ram

e i

s a

rect

an

gle

?

2

. B

OO

KS

HE

LV

ES

A

book

shelf

con

sist

s of

two v

ert

ical

pla

nk

s w

ith

fiv

e h

ori

zon

tal

shelv

es.

Are

each

of

the

fou

r se

ctio

ns

for

book

s re

ctan

gle

s? E

xp

lain

.

3

. LA

ND

SC

AP

ING

A

lan

dsc

ap

er

is

mark

ing o

ff t

he c

orn

ers

of

a r

ect

an

gu

lar

plo

t of

lan

d.

Th

ree o

f th

e c

orn

ers

are

in

p

lace

as

show

n.

Wh

at

are

th

e c

oord

inate

s of

the f

ou

rth

co

rner?

4

. S

WIM

MIN

G P

OO

LS

A

nto

nio

is

desi

gn

ing a

sw

imm

ing p

ool

on

a

coord

inate

gri

d.

Is i

t a r

ect

an

gle

? E

xp

lain

.

5

. P

AT

TE

RN

SV

ero

nic

a m

ad

e t

he p

att

ern

sh

ow

n o

ut

of

7 r

ect

an

gle

s w

ith

fou

r equ

al

sid

es.

Th

e s

ide l

en

gth

of

each

re

ctan

gle

is

wri

tten

in

sid

e t

he r

ect

an

gle

.

a.

How

man

y r

ect

an

gle

s ca

n b

e f

orm

ed

u

sin

g t

he l

ines

in t

his

fig

ure

?

b.

If V

ero

nic

a w

an

ted

to e

xte

nd

her

patt

ern

by a

dd

ing a

noth

er

rect

an

gle

w

ith

4 e

qu

al

sid

es

to m

ak

e a

larg

er

rect

an

gle

, w

hat

are

th

e p

oss

ible

si

de l

en

gth

s of

rect

an

gle

s th

at

she

can

ad

d?

Wo

rd

Pro

ble

m P

racti

ce

Recta

ng

les

1 1

2 3

5

85y

xO

5

5y

xO

–5

–5

AB

DC

6 -4

T

hey s

ho

uld

be e

qu

al.

Y

es;

each

of

the f

ou

r an

gle

s o

f

each

recta

ng

le i

s c

reate

d b

y a

str

aig

ht

ho

rizo

nta

l lin

e a

nd

a

str

aig

ht

vert

ical

lin

e,

so

each

has f

ou

r 90

° an

gle

s.

(-6,

3)

Y

es.

Sam

ple

an

sw

er:

Th

e s

lop

e o

f

the l

on

g s

ides i

s 3

an

d t

he s

lop

e

o

f th

e s

ho

rt s

ides i

s -

1

−

3 ,

so

each

p

air

of

sid

es i

s p

ara

llel. A

lso

, th

e

lon

g a

nd

sh

ort

sid

es h

ave s

lop

es

that

are

perp

en

dic

ula

r to

each

oth

er,

makin

g t

he f

ou

r co

rners

rig

ht

an

gle

s.

11

8 a

nd

13 u

nit

s


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

28

G

len

co

e G

eo

me

try

Co

nsta

nt

Peri

mete

rD

ou

gla

s w

an

ts t

o f

en

ce a

rect

an

gu

lar

regio

n o

f h

is b

ack

yard

for

his

dog.

He b

ou

gh

t 200 f

eet

of

fen

ce.

1. C

om

ple

te t

he t

able

to s

how

th

e

dim

en

sion

s of

five d

iffe

ren

t re

ctan

gu

lar

pen

s th

at

wou

ld u

se t

he e

nti

re 2

00 f

eet

of

fen

ce.

Th

en

fin

d t

he a

rea o

f each

re

ctan

gu

lar

pen

.

2. D

o a

ll f

ive o

f th

e r

ect

an

gu

lar

pen

s h

ave

the s

am

e a

rea?

If n

ot,

wh

ich

on

e h

as

the l

arg

er

are

a?

3. W

rite

a r

ule

for

fin

din

g t

he d

imen

sion

s of

a r

ect

an

gle

wit

h

the l

arg

est

poss

ible

are

a f

or

a g

iven

peri

mete

r.

4. L

et x r

ep

rese

nt

the l

en

gth

of

a r

ect

an

gle

an

d y

th

e w

idth

. W

rite

th

e f

orm

ula

for

all

rect

an

gle

s w

ith

a p

eri

mete

r of

200.

Th

en

gra

ph

th

is r

ela

tion

ship

on

th

e c

oord

inate

pla

ne

at

the r

igh

t.

Ju

lio r

ead

th

at

a d

og t

he s

ize o

f h

is n

ew

pet,

Ben

nie

, sh

ou

ld h

ave a

t le

ast

100 s

qu

are

feet

in

his

pen

. B

efo

re g

oin

g t

o t

he s

tore

to b

uy f

en

ce,

Ju

lio m

ad

e a

table

to d

ete

rmin

e t

he

dim

en

sion

s fo

r B

en

nie

’s r

ect

an

gu

lar

pen

.

5. C

om

ple

te t

he t

able

to f

ind

fiv

e p

oss

ible

d

imen

sion

s of

a r

ect

an

gu

lar

fen

ced

are

a o

f 100 s

qu

are

feet.

6. Ju

lio w

an

ts t

o s

ave m

on

ey b

y p

urc

hasi

ng

the l

east

nu

mber

feet

of

fen

cin

g t

o e

ncl

ose

th

e

100 s

qu

are

feet.

Wh

at

wil

l be t

he d

imen

sion

s of

the c

om

ple

ted

pen

?

7. W

rite

a r

ule

for

fin

din

g t

he d

imen

sion

s of

a r

ect

an

gle

wit

h

the l

east

poss

ible

peri

mete

r fo

r a g

iven

are

a.

8. F

or

len

gth

x a

nd

wid

th y

, w

rite

a f

orm

ula

for

the a

rea o

f a

rect

an

gle

wit

h a

n a

rea o

f 100 s

qu

are

feet.

Th

en

gra

ph

th

e

form

ula

.

100

100

y

xOy

xO

100

100

6-4

Pe

rim

ete

rL

en

gth

Wid

thA

rea

20

08

0

20

07

0

20

06

0

20

05

0

20

04

5

Are

aL

en

gth

Wid

thH

ow

mu

ch

fen

ce

to

bu

y

10

0

10

0

10

0

10

0

10

0

En

rich

men

t

N

o,

the 5

0 ×

50 p

en

has t

he l

arg

est

are

a.

T

he r

ecta

ng

le w

ith

th

e l

arg

est

are

a f

or

a g

iven

peri

mete

r is

a s

qu

are

.

2

x +

2y =

200 o

r x +

y =

100

S

ee t

ab

le f

or

sam

ple

an

sw

ers

.

1

0 f

t ×

10 f

t

T

he r

ecta

ng

le w

ith

th

e l

east

po

ssib

le p

eri

mete

r fo

r

a g

iven

are

a i

s a

sq

uare

.

x

y=

10

0

11

00

202

250

104

425

58

520

50

10

10

40

20

1600

30

2100

40

2400

50

2500

55

2475

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

29

G

len

co

e G

eo

me

try

Gra

ph

ing C

alc

ula

tor

Act

ivit

y

TI-

Nsp

ire:

Exp

lori

ng

Recta

ng

les

A q

uad

rila

tera

l w

ith

fou

r ri

gh

t an

gle

s is

a r

ecta

ng

le.

Th

e T

I-N

spir

e c

an

be u

sed

to e

xp

lore

so

me o

f th

e c

hara

cteri

stic

s of

a r

ect

an

gle

. U

se t

he f

oll

ow

ing s

tep

s to

dra

w a

rect

an

gle

.

Ste

p 1

S

et

up

th

e c

alc

ula

tor

in t

he c

orr

ect

mod

e.

• C

hoose

Gra

ph

s &

Ge

om

etr

y f

rom

th

e H

om

e M

en

u.

• F

rom

th

e V

iew

men

u, ch

oose

4:

Hid

e A

xis

Ste

p 2

D

raw

th

e r

ect

an

gle

• F

rom

th

e 8

: S

ha

pe

s m

en

u c

hoose

3:

Re

cta

ng

le.

• C

lick

on

ce t

o d

efi

ne t

he c

orn

er

of

the r

ect

an

gle

. T

hen

move a

nd

cli

ck a

gain

. T

he

sid

e o

f th

e r

ect

an

gle

is

now

defi

ned

. M

ove p

erp

en

dic

ula

rly t

o d

raw

th

e r

ect

an

gle

. C

lick

to a

nch

or

the s

hap

e.

Ste

p 3

M

easu

re t

he l

en

gth

s of

the s

ides

of

the r

ect

an

gle

.

• F

rom

th

e 7

: M

ea

su

re

me

nt

men

u c

hoose

1:

Le

ng

th (

Note

th

at

wh

en

you

scr

oll

over

the r

ect

an

gle

, th

e v

alu

e n

ow

sh

ow

n i

s th

e p

eri

mete

r of

the r

ect

an

gle

.)

• S

ele

ct e

ach

en

dp

oin

t of

a s

egm

en

t of

the r

ect

an

gle

. T

hen

cli

ck o

r p

ress

En

ter

to

an

chor

the l

en

gth

of

the s

egm

en

t in

th

e w

ork

are

a.

• R

ep

eat

for

the o

ther

sid

es

of

the r

ect

an

gle

.

Exercis

es

1

. W

hat

ap

pears

to b

e t

rue a

bou

t th

e o

pp

osi

te s

ides

of

the r

ect

an

gle

?

2

. D

raw

th

e d

iagon

als

of

the r

ect

an

gle

usi

ng 5

: S

eg

me

nt

from

th

e 6

: P

oin

ts a

nd

Lin

es

Men

u.

Cli

ck o

n t

wo o

pp

osi

te v

ert

ices

to d

raw

th

e d

iagon

al.

Rep

eat

to d

raw

th

e o

ther

dia

gon

al.

a.

Measu

re e

ach

dia

gon

al

usi

ng t

he m

easu

rem

en

t to

ol.

Wh

at

do y

ou

obse

rve?

b.

Wh

at

is t

rue a

bou

t th

e t

rian

gle

s fo

rmed

by t

he s

ides

of

the r

ect

an

gle

an

d a

dia

gon

al?

Ju

stif

y y

ou

r co

ncl

usi

on

.

3

. P

ress

Cle

ar t

hre

e t

imes

an

d s

ele

ct Y

es t

o c

lear

the s

creen

. R

ep

eat

the s

tep

s an

d d

raw

an

oth

er

rect

an

gle

. D

o t

he r

ela

tion

ship

s th

at

you

fou

nd

for

the f

irst

rect

an

gle

you

dre

w

hold

tru

e f

or

this

rect

an

gle

?

6-4

T

he o

pp

osit

e s

ides o

f a r

ecta

ng

le a

re c

on

gru

en

t.

b

. T

he t

rian

gle

s a

re c

on

gru

en

t b

y S

SS

.

Y

es,

the o

pp

osit

e s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

co

ng

ruen

t.

a

. T

he d

iag

on

als

of

a r

ecta

ng

le a

re c

on

gru

en

t.


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

30

G

len

co

e G

eo

me

try

Geo

mete

r’s

Sk

etc

hp

ad

Act

ivit

y

Exp

lori

ng

Recta

ng

les

A q

uad

rila

tera

l w

ith

fou

r ri

gh

t an

gle

s is

a r

ecta

ng

le.

Th

e G

eom

ete

r’s

Sk

etc

hp

ad

is

a u

sefu

l

tool

for

exp

lori

ng s

om

e o

f th

e c

hara

cteri

stic

s of

a r

ect

an

gle

. U

se t

he f

oll

ow

ing s

tep

s to

dra

w

a r

ect

an

gle

.

Ste

p 1

U

se t

he L

ine t

ool

to d

raw

a l

ine a

nyw

here

on

th

e s

creen

.

Ste

p 2

U

se t

he P

oin

t to

ol

to d

raw

a p

oin

t th

at

is

not

on

th

e l

ine.

To d

raw

a l

ine p

erp

en

dic

ula

r

to t

he f

irst

lin

e y

ou

dre

w,

sele

ct t

he f

irst

lin

e

an

d t

he p

oin

t. T

hen

ch

oose

Pe

rp

en

dic

ula

r

Lin

e f

rom

th

e C

on

str

uct

men

u.

Ste

p 3

U

se t

he P

oin

t to

ol

to d

raw

a p

oin

t th

at

is

not

on

eit

her

of

the l

ines

you

have d

raw

n.

Rep

eat

the p

roce

du

re i

n S

tep

2 t

o d

raw

lin

es

perp

en

dic

ula

r to

th

e t

wo l

ines

you

have d

raw

n.

A r

ect

an

gle

is

form

ed

by t

he s

egm

en

ts w

hose

en

dp

oin

ts a

re t

he p

oin

ts o

f in

ters

ect

ion

of

the

lin

es.

Exerc

ises

Use

th

e m

ea

su

rin

g c

ap

ab

ilit

ies o

f T

he

Ge

om

ete

r’s

Sk

etc

hp

ad

to

ex

plo

re

the

ch

ara

cte

ris

tics o

f a

re

cta

ng

le.

1

. W

hat

ap

pears

to b

e t

rue a

bou

t th

e o

pp

osi

te s

ides

of

the r

ect

an

gle

th

at

you

dre

w?

Mak

e a

con

ject

ure

an

d t

hen

measu

re e

ach

sid

e t

o c

heck

you

r co

nje

ctu

re.

2

. D

raw

th

e d

iagon

als

of

the r

ect

an

gle

by u

sin

g t

he S

ele

ctio

n A

rrow

tool

to c

hoose

tw

o

op

posi

te v

ert

ices.

Th

en

ch

oose

Se

gm

en

t fr

om

th

e C

on

str

uct

men

u t

o d

raw

th

e

dia

gon

al.

Rep

eat

to d

raw

th

e o

ther

dia

gon

al.

a.

Measu

re e

ach

dia

gon

al.

Wh

at

do y

ou

obse

rve?

b.

Wh

at

is t

rue a

bou

t th

e t

rian

gle

s fo

rmed

by t

he s

ides

of

the r

ect

an

gle

an

d a

dia

gon

al?

Ju

stif

y y

ou

r co

ncl

usi

on

.

3

. C

hoose

Ne

w S

ke

tch

fro

m t

he F

ile

men

u a

nd

foll

ow

ste

ps

1–3 t

o d

raw

an

oth

er

rect

an

gle

. D

o t

he r

ela

tion

ship

s you

fou

nd

for

the f

irst

rect

an

gle

you

dre

w h

old

tru

e f

or

this

rect

an

gle

als

o?

6 -4

T

he o

pp

osit

e s

ides o

f a r

ecta

ng

le a

re c

on

gru

en

t.

Th

e d

iag

on

als

of

a r

ecta

ng

le a

re c

on

gru

en

t.

Th

e t

rian

gle

s a

re c

on

gru

en

t b

y S

SS

.

Y

es,

the o

pp

osit

e s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

co

ng

ruen

t.


NA

ME

DA

TE

PE

RIO

D

Lesson 6-5

Ch

ap

ter

6

31

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Rh

om

bi

an

d S

qu

are

s

Pro

pe

rtie

s o

f R

ho

mb

i a

nd

Sq

ua

res

A r

ho

mb

us i

s a q

uad

rila

tera

l w

ith

fou

r co

ngru

en

t si

des.

Op

posi

te s

ides

are

con

gru

en

t, s

o a

rh

om

bu

s is

als

o a

para

llelo

gra

m a

nd

has

all

of

the p

rop

ert

ies

of

a p

ara

llelo

gra

m.

Rh

om

bi

als

o h

ave t

he f

oll

ow

ing p

rop

ert

ies.

A s

qu

are

is

a p

ara

llelo

gra

m w

ith

fou

r co

ngru

en

t si

des

an

d f

ou

r

con

gru

en

t an

gle

s. A

squ

are

is

both

a r

ect

an

gle

an

d a

rh

om

bu

s;

there

fore

, all

pro

pert

ies

of

para

llelo

gra

ms,

rect

an

gle

s, a

nd

rh

om

bi

ap

ply

to s

qu

are

s.

In

rh

om

bu

s ABCD

, m

∠BAC

= 3

2.

Fin

d t

he

me

asu

re

of

ea

ch

nu

mb

ere

d a

ng

le.

ABCD

is

a r

hom

bu

s, s

o t

he d

iagon

als

are

perp

en

dic

ula

r an

d △

ABE

is a

rig

ht

tria

ngle

. T

hu

s m

∠4 =

90 a

nd

m∠

1 =

90 -

32 o

r 58.

Th

e

dia

gon

als

in

a r

hom

bu

s bis

ect

th

e v

ert

ex a

ngle

s, s

o m

∠1 =

m∠

2.

Th

us,

m∠

2 =

58

.

A r

hom

bu

s is

a p

ara

llelo

gra

m,

so t

he o

pp

osi

te s

ides

are

para

llel.

∠BAC

an

d ∠

3 a

re

alt

ern

ate

in

teri

or

an

gle

s fo

r p

ara

llel

lin

es,

so m

∠3 =

32

.

Exerc

ises

Qu

ad

ril

ate

ra

l ABCD

is a

rh

om

bu

s.

Fin

d e

ach

va

lue

or m

ea

su

re

.

1

. If

m∠ABD

= 6

0,

fin

d m

∠BDC

. 2. If

AE

= 8

, fi

nd

AC

.

3

. If

AB

= 2

6 a

nd

BD

= 2

0,

fin

d AE

. 4. F

ind

m∠CEB

.

5

. If

m∠CBD

= 5

8,

fin

d m

∠ACB

. 6. If

AE

= 3x -

1 a

nd

AC

= 1

6,

fin

d x

.

7. If

m∠CDB

= 6y a

nd

m∠ACB

= 2y +

10,

fin

d y

.

8. If

AD

= 2x +

4 a

nd

CD

= 4x -

4,

fin

d x

.

MO

HR

B

CA

D

E

B

32°

12

3

4

C

AD

EB

6 -5

Th

e d

iag

on

als

are

pe

rpe

nd

icu

lar.

−−

−

MH

⊥ −

−−

RO

Ea

ch

dia

go

na

l b

ise

cts

a p

air o

f o

pp

osite

an

gle

s.

−−

−

MH

bis

ects

∠R

MO

an

d ∠

RH

O.

−−

−

RO

bis

ects

∠M

RH

an

d ∠

MO

H.

Exam

ple

60

16

24

90

32

3

10

4


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

32

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Rh

om

bi

an

d S

qu

are

s

Co

nd

itio

ns

for

Rh

om

bi

an

d S

qu

are

s T

he t

heore

ms

belo

w c

an

help

you

pro

ve t

hat

a p

ara

llelo

gra

m i

s a r

ect

an

gle

, rh

om

bu

s, o

r sq

uare

.

If t

he d

iagon

als

of

a p

ara

llelo

gra

m a

re p

erp

en

dic

ula

r, t

hen

th

e p

ara

llelo

gra

m i

s a r

hom

bu

s.

If o

ne d

iagon

al

of

a p

ara

llelo

gra

m b

isect

s a p

air

of

op

posi

te a

ngle

s, t

hen

th

e p

ara

llelo

gra

m

is a

rh

om

bu

s.

If o

ne p

air

of

con

secu

tive s

ides

of

a p

ara

llelo

gra

m a

re c

on

gru

en

t, t

he p

ara

llelo

gra

m i

s a

rhom

bu

s.

If a

qu

ad

rila

tera

l is

both

a r

ect

an

gle

an

d a

rh

om

bu

s, t

hen

it

is a

squ

are

.

D

ete

rm

ine

wh

eth

er p

ara

lle

log

ra

m ABCD

wit

h

ve

rti

ce

s A

(−3

, −

3),

B(1

, 1

), C

(5,

−3

), D

(1,

−7

) is

a rhombus,

a rectangle

, o

r a

square.

AC

= √

""

""

""

""

""

(-3 -

5)2

+ (

(-3 -

(-

3))

2 =

√ "

"

64 =

8

BD

= √

""

""

""

""

(1

-1)2

+ (

-7 -

1)2

= √

""

64 =

8

Th

e d

iagon

als

are

th

e s

am

e l

en

gth

; th

e f

igu

re i

s a r

ect

an

gle

.

Slo

pe o

f −

−

AC

=

-3 -

(-

3)

−

-3 -

5

=

0

−

-8 =

8

Th

e l

ine i

s h

ori

zon

tal.

Slo

pe o

f −−

− B

D =

1 -

(-

7)

−

1 -

1

=

8

−

0 =

un

def

ined

T

he l

ine i

s vert

ical.

Sin

ce a

hori

zon

tal

an

d v

ert

ical

lin

e a

re p

erp

en

dic

ula

r, t

he d

iagon

als

are

perp

en

dic

ula

r.

Para

llelo

gra

m A

BC

D i

s a s

qu

are

wh

ich

is

als

o a

rh

om

bu

s an

d a

rect

an

gle

.

Exerc

ises

Giv

en

ea

ch

se

t o

f v

erti

ce

s,

de

term

ine

wh

eth

er !

ABCD

is a

rhombus, rectangle

, o

r

square.

Lis

t a

ll t

ha

t a

pp

ly.

Ex

pla

in.

1. A

(0,

2),

B(2

, 4),

C(4

, 2),

D(2

, 0)

2. A

(-2,

1),

B(-

1,

3),

C(3

, 1),

D(2

, -

1)

3. A

(-2,

-1),

B(0

, 2),

C(2

, -

1),

D(0

, -

4)

4. A

(-3,

0),

B(-

1,

3),

C(5

, -

1),

D(3

, -

4)

5

. P

RO

OF W

rite

a t

wo-c

olu

mn

pro

of.

Giv

en

: P

ara

llelo

gra

m R

ST

U.

−−

RS

$ −

−

ST

Pro

ve

: R

ST

U i

s a r

hom

bu

s.

Sta

tem

en

ts

Re

as

on

s

1.

RS

TU

is

a p

ara

lle

log

ram

1

. G

ive

n

−

−

RS

$ −

−

ST

2. −

−

RS

$ −

−

UT , −

−

RU

$ −

−

ST

2.

De

fin

itio

n o

f a

pa

rall

elo

gra

m

3. −

−

UT $

−−

RS

$ −

−

ST $

−−

RU

3

. S

ub

sti

tuti

on

4.

RS

TU

is

a r

ho

mb

us

4

. D

efi

nit

ion

of

a r

ho

mb

us

6 -5 Exam

ple

y

x

R

ec

tan

gle

, rh

om

bu

s,

sq

ua

re;

the

fo

ur

sid

es

are

$ a

nd

co

ns

ec

uti

ve

s

ide

s a

re ⊥

.

R

ec

tan

gle

; c

on

se

cu

tiv

e s

ide

s a

re ⊥

.

R

ho

mb

us

; th

e f

ou

r s

ide

s a

re $

a

nd

co

ns

ec

uti

ve

sid

es

are

no

t ⊥

.

R

ec

tan

gle

; c

on

se

cu

tiv

e s

ide

s a

re ⊥

.

Lesson 6-5


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

33

G

len

co

e G

eo

me

try

Sk

ills

Pra

ctic

e

Rh

om

bi

an

d S

qu

are

s

ALG

EB

RA

Q

ua

dril

ate

ra

l DKLM

is a

rh

om

bu

s.

1

. If

DK

= 8

, fi

nd

KL

.

2

. If

m∠

DM

L=

82 f

ind

m∠

DK

M.

3

. If

m∠

KA

L=

2x

– 8,

fin

d x

.

4

. If

DA

= 4

x a

nd

AL

= 5

x–

3,

fin

d D

L.

5

. If

DA

= 4

x a

nd

AL

= 5

x–

3,

fin

d A

D.

6

. If

DM

= 5

y+

2 a

nd

DK

= 3

y+

6,

fin

d K

L.

7.P

RO

OF

Wri

te a

tw

o-c

olu

mn

pro

of.

Giv

en

:R

ST

U i

s a p

ara

llelo

gra

m.

−−−

RX

$−

−T

X $

−−

SX

$−−

−U

X

Pro

ve

:R

ST

U i

s a r

ect

an

gle

.P

ro

of

S

tate

me

nts

Re

aso

ns

1

.R

STU

is a

para

llelo

gra

m.

1.

Giv

en

−−

RX

$ −

−TX

$ −

−S

X $

−−

UX

2.

RX

= T

X =

SX

= U

X2.

Defi

nit

ion

of

co

ng

ruen

t seg

men

ts

3.

RX

+ X

T =

RT

, S

X +

XU

= S

U3.

Seg

Ad

d P

ostu

late

4.

RX

+X

T=

SU

4.

Su

bsti

tuti

on

Pro

pert

y

5.

RT =

SU

5.

Tra

nsit

ive P

rop

ert

y

6.

−−

RT $

−−

SU

6.

Defi

nit

ion

of

$ s

eg

men

t

7.

RS

TU

is a

recta

ng

le.

7.

If d

iag

on

als

$,

the f

igu

re i

s a

re

cta

ng

le.

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ive

n e

ach

se

t o

f v

erti

ce

s,

de

term

ine

wh

eth

er !

QRST

is a

rhombus,

a rectangle

, o

r a

square.

Lis

t a

ll t

ha

t a

pp

ly.

Ex

pla

in.

8. Q

(3,

5),

R(3

, 1),

S(-

1,

1),

T(-

1,

5)

9

. Q

(-5,

12),

R(5

, 12),

S(-

1,

4),

T(-

11,

4)

10

. Q

(-6,

-1),

R(4

, -

6),

S(2

, 5),

T(-

8,

10)

11

. Q

(2,

-4),

R(-

6,

-8),

S(-

10,

2),

T(-

2,

6)

ML

AD

K

6 -5

8

R

ho

mb

us,

recta

ng

le,

sq

uare

; all s

ides a

re c

on

gru

en

t an

d t

he

dia

go

nals

are

perp

en

dic

ula

r an

d c

on

gru

en

t.

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

N

on

e;

op

po

sit

e s

ides a

re c

on

gru

en

t, b

ut

the d

iag

on

als

are

neit

her

co

ng

ruen

t n

or

perp

en

dic

ula

r.

41

49

24

12

12


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

34

G

len

co

e G

eo

me

try

Practi

ce

Rh

om

bi

an

d S

qu

are

s

PRYZ

is a

rh

om

bu

s.

If RK

= 5

, RY

= 1

3 a

nd

m∠YRZ

= 6

7,

fin

d e

ach

me

asu

re

.

1

. K

Y

2

. P

K

3

. m∠

YK

Z

4

. m∠

PZ

R

MNPQ

is a

rh

om

bu

s.

If PQ

= 3

√ #

2

an

d AP

= 3

, fi

nd

ea

ch

me

asu

re

.

5

. A

Q

6

. m∠

AP

Q

7

. m∠

MN

P

8

. P

M

CO

OR

DIN

AT

E G

EO

ME

TR

Y G

ive

n e

ach

se

t o

f v

erti

ce

s,

de

term

ine

wh

eth

er $

BEFG

is a

rhombus,

a rectangle

, o

r a

square.

Lis

t a

ll t

ha

t a

pp

ly.

Ex

pla

in.

9. B

(-9,

1),

E(2

, 3),

F(1

2, -

2),

G(1

, -

4)

10

. B

(1,

3),

E(7

, -

3),

F(1

, -

9),

G(-

5, -

3)

11

. B

(-4, -

5),

E(1

, -

5),

F(-

2, -

1),

G(-

7, -

1)

12

. T

ES

SE

LLA

TIO

NS

T

he f

igu

re i

s an

exam

ple

of

a t

ess

ell

ati

on

. U

se a

ru

ler

or

pro

tract

or

to m

easu

re t

he s

hap

es

an

d t

hen

nam

e t

he q

uad

rila

tera

ls

use

d t

o f

orm

th

e f

igu

re.

K

P

YR

Z

A

N MQP

6 -5

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

R

ho

mb

us,

recta

ng

le,

sq

uare

; all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r an

d c

on

gru

en

t.

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

T

he f

igu

re c

on

sis

ts o

f 6 c

on

gru

en

t rh

om

bi.

12

12

90

67

45 90

63


NA

ME

DA

TE

PE

RIO

D

Lesson 6-5

Ch

ap

ter

6

35

G

len

co

e G

eo

me

try

1. T

RA

Y R

AC

KS

A

tra

y r

ack

look

s li

ke a

p

ara

llelo

gra

m f

rom

th

e s

ide.

Th

e l

evels

fo

r th

e t

rays

are

even

ly s

pace

d.

Wh

at

two l

abele

d p

oin

ts f

orm

a r

hom

bu

s w

ith

base

AA' ?

2. S

LIC

ING

C

harl

es

cuts

a r

hom

bu

s alo

ng

both

dia

gon

als

. H

e e

nd

s u

p w

ith

fou

r co

ngru

en

t tr

ian

gle

s. C

lass

ify t

hese

tr

ian

gle

s as

acu

te,

obtu

se,

or

righ

t.

3. W

IND

OW

S T

he e

dges

of

a w

ind

ow

are

d

raw

n i

n t

he c

oord

inate

pla

ne.

Dete

rmin

e w

heth

er

the w

ind

ow

is

a

squ

are

or

a r

hom

bu

s.

4.S

QU

AR

ES

M

ack

en

zie

cu

t a s

qu

are

alo

ng i

ts d

iagon

als

to g

et

fou

r co

ngru

en

t ri

gh

t tr

ian

gle

s. S

he t

hen

join

ed

tw

o o

f th

em

alo

ng t

heir

lon

g s

ides.

Sh

ow

th

at

the r

esu

ltin

g s

hap

e i

s a s

qu

are

.

5. D

ES

IGN

T

ati

an

na m

ad

e t

he d

esi

gn

sh

ow

n.

Sh

e u

sed

32 c

on

gru

en

t rh

om

bi

to

create

th

e f

low

er-

lik

e d

esi

gn

at

each

co

rner.

a.

Wh

at

are

th

e a

ngle

s of

the c

orn

er

rhom

bi?

b.

Wh

at

kin

ds

of

qu

ad

rila

tera

ls a

re t

he

dott

ed

an

d c

heck

ere

d f

igu

res?

Wo

rd

Pro

ble

m P

racti

ce

Rh

om

bi

an

d S

qu

are

s

y

xO

AB

C

D

E

F

G

H

B,C,D,E,F,G,H,

A20 inches

AB

= 4

in

ch

es

,

6 -5

F a

nd

F′

rig

ht

tria

ng

les

rho

mb

us

Sam

ple

an

sw

er:

Th

e d

iag

on

als

of

a s

qu

are

cre

ate

fo

ur

co

ng

ruen

t

rig

ht

iso

scele

s t

rian

gle

s.

Wh

en

two

co

ng

ruen

t ri

gh

t is

oscele

s

tria

ng

les a

re j

oin

ed

alo

ng

th

eir

hyp

ote

nu

ses,

the r

esu

lt i

s a

qu

ad

rila

tera

l w

ith

4 e

qu

al

sid

es

an

d 4

rig

ht

an

gle

co

rners

(becau

se 4

5 +

45 =

90),

makin

g

it a

sq

uare

. 45,

135,

45,

135

Th

ey a

re a

ll s

qu

are

s.


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

36

G

len

co

e G

eo

me

try

En

rich

men

t

Cre

ati

ng

Pyth

ag

ore

an

Pu

zzle

s

By d

raw

ing t

wo s

qu

are

s an

d c

utt

ing t

hem

in

a c

ert

ain

way,

you

ca

n m

ak

e a

pu

zzle

th

at

dem

on

stra

tes

the P

yth

agore

an

Th

eore

m.

A s

am

ple

pu

zzle

is

show

n.

You

can

cre

ate

you

r ow

n p

uzzle

by

foll

ow

ing t

he i

nst

ruct

ion

s belo

w.

1

. C

are

full

y c

on

stru

ct a

squ

are

an

d l

abel

the l

en

gth

of

a s

ide a

s a

. T

hen

con

stru

ct a

sm

all

er

squ

are

to

the r

igh

t of

it a

nd

label

the l

en

gth

of

a s

ide a

s b,

as

show

n i

n t

he f

igu

re a

bove.

Th

e b

ase

s sh

ou

ld

be a

dja

cen

t an

d c

oll

inear.

2

. M

ark

a p

oin

t X

th

at

is b

un

its

from

th

e l

eft

ed

ge

of

the l

arg

er

squ

are

. T

hen

dra

w t

he s

egm

en

ts

from

th

e u

pp

er

left

corn

er

of

the l

arg

er

squ

are

to

poin

t X

, an

d f

rom

poin

t X

to t

he u

pp

er

righ

t co

rner

of

the s

mall

er

squ

are

.

3

. C

ut

ou

t an

d r

earr

an

ge y

ou

r fi

ve p

iece

s to

form

a

larg

er

squ

are

. D

raw

a d

iagra

m t

o s

how

you

r an

swer.

4

. V

eri

fy t

hat

the l

en

gth

of

each

sid

e i

s equ

al

to

√ "

""

a2 +

b2 .

a

a

bX

b

b

6 -5

See s

tud

en

ts’

wo

rk.

A s

am

ple

an

sw

er

is s

ho

wn

.

Lesson 6-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

37

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Tra

pezo

ids a

nd

Kit

es

Pro

pe

rtie

s o

f T

rap

ezo

ids

A t

ra

pe

zo

id i

s a q

uad

rila

tera

l w

ith

exact

ly o

ne p

air

of

para

llel

sid

es.

Th

e m

idse

gm

en

t or

me

dia

n

of

a t

rap

ezoid

is

the s

egm

en

t th

at

con

nect

s th

e m

idp

oin

ts o

f th

e l

egs

of

the t

rap

ezoid

. It

s m

easu

re i

s equ

al

to o

ne-h

alf

th

e s

um

of

the

len

gth

s of

the b

ase

s. I

f th

e l

egs

are

con

gru

en

t, t

he t

rap

ezoid

is

an

iso

sce

les t

ra

pe

zo

id.

In a

n i

sosc

ele

s tr

ap

ezoid

both

pair

s of

ba

se

an

gle

s a

re c

on

gru

en

t an

d t

he d

iagon

als

are

con

gru

en

t.

T

he

ve

rti

ce

s o

f ABCD

are

A(-

3,

-1

), B

(-1

, 3

),

C(2

, 3

), a

nd

D(4

, -

1).

Sh

ow

th

at ABCD

is a

tra

pe

zo

id a

nd

d

ete

rm

ine

wh

eth

er i

t is

an

iso

sce

les

tra

pe

zo

id.

slop

e o

f −

−

AB

=

3 -

(-

1)

−

-1 -

(-

3) =

4

−

2 =

2

slop

e o

f −−

− A

D =

-1 -

(-

1)

−

4 -

(-

3)

= 0

−

7 =

0

slop

e o

f −−

− B

C =

3 -

3

−

2 -

(-

1)

= 0

−

3 =

0

slop

e o

f −−

− C

D =

-1 -

3

−

4 -

2 =

-4

−

2 =

-2

Exact

ly t

wo s

ides

are

para

llel,

−−−

AD

an

d −−

− B

C ,

so A

BC

D i

s a t

rap

ezoid

. A

B =

CD

, so

AB

CD

is

an

iso

scele

s tr

ap

ezoid

.

Exerc

ises

Fin

d e

ach

me

asu

re

.

1

. m

∠D

55

2

. m

∠L

140

125°

5

5

40°

CO

OR

DIN

AT

E G

EO

ME

TR

Y F

or e

ach

qu

ad

ril

ate

ra

l w

ith

th

e g

ive

n v

erti

ce

s,

ve

rif

y

tha

t th

e q

ua

dril

ate

ra

l is

a t

ra

pe

zo

id a

nd

de

term

ine

wh

eth

er t

he

fig

ure

is a

n

iso

sce

les t

ra

pe

zo

id.

3

. A

(−1,

1),

B(3

, 2),

C(1

,−2),

D(−

2,

−1)

4. J

(1,

3),

K(3

, 1),

L(3

, −

2),

M(−

2,

3)

−

−

AD

‖ −

−

BC

, −

−

AB

∦ −

−

CD

, A

BC

D i

s a

−−

JK

‖ −

−

LM

, −

−

JM

∦ −

−

KL

, JK

LM

is a

n

trap

ezo

id b

ut

no

t an

iso

scele

s

iso

scele

s t

rap

ezo

id b

ecau

se

trap

ezo

id b

ecau

se A

B =

√ %

%

17 ,

J

M =

KL =

3.

CD

= √

%%

10 .

Fo

r t

ra

pe

zo

id HJKL

, M

an

d N

are

th

e m

idp

oin

ts o

f th

e l

eg

s.

5. I

f H

J =

32 a

nd

LK

= 6

0,

fin

d M

N.

46

6. If

HJ

= 1

8 a

nd

MN

= 2

8,

fin

d L

K.

38

T

RU

bas

e

bas

e

Sleg

x

y

O

B

DA

C

6-6

STU

R is a

n isoscele

s t

rapezoid

.

−−

SR

& −

−

TU

; ∠

R &

∠U

, ∠

S &

∠T

Exam

ple

AB

=

√ "

""

""

""

""

"

(-3 -

(-

1))

2 +

(-

1 -

3)2

=

√

""

"

4 +

16

= √

""

20 =

2 √

"

5

CD

=

√ "

""

""

""

""

(2

- 4

)2 +

(3 -

(-

1))

2

=

√ "

""

4 +

16 =

√ "

"

20 =

2 √

"

5

LK

NM

HJ


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

38

G

len

co

e G

eo

me

try

Pro

pe

rtie

s o

f K

ite

s A

kit

e i

s a q

uad

rila

tera

l w

ith

exact

ly t

wo p

air

s of

con

secu

tive

con

gru

en

t si

des.

Un

lik

e a

para

llelo

gra

m,

the o

pp

osi

te s

ides

of

a k

ite a

re n

ot

con

gru

en

t or

para

llel.

Th

e d

iagon

als

of

a k

ite a

re p

erp

en

dic

ula

r.

For

kit

e R

MN

P,

−−

−

MP

⊥ −

−−

RN

In a

kit

e,

exact

ly o

ne p

air

of

op

posi

te a

ngle

s is

con

gru

en

t.

For

kit

e R

MN

P,

∠M

$ ∠

P

If

WX

YZ

is a

kit

e,

fin

d m

∠Z

.

Th

e m

easu

res

of

∠Y

an

d ∠

W a

re n

ot

con

gru

en

t, s

o ∠

X $

∠Z

.

m∠

X +

m∠

Y +

m∠

Z +

m∠

W =

360

m∠

X +

60

+ m

∠Z

+ 8

0 =

360

m∠

X +

m∠

Z =

220

m∠

X =

110, m

∠Z

= 1

10

If

AB

CD

is a

kit

e,

fin

d B

C.

Th

e d

iagon

als

of

a k

ite a

re p

erp

en

dic

ula

r. U

se t

he

Pyth

agore

an

Th

eore

m t

o f

ind

th

e m

issi

ng l

en

gth

.

BP

2 +

PC

2 =

BC

2

52 +

12

2 =

BC

2

169 =

BC

2

13 =

BC

Exerc

ises

If G

HJ

K i

s a

kit

e,

fin

d e

ach

me

asu

re

.

1. F

ind

m∠

JR

K.

90

2

. If

RJ

= 3

an

d R

K =

10,

fin

d J

K.

√ #

#

109

3

. If

m∠

GH

J =

90 a

nd

m∠

GK

J =

11

0,

fin

d m

∠H

GK

.

80

4

. If

HJ

= 7

, fi

nd

HG

.

7

5

. If

HG

= 7

an

d G

R =

5,

fin

d H

R.

√ #

#

24 =

2 √

#

6

6

. If

m∠

GH

J =

52 a

nd

m∠

GK

J =

95,

fin

d m

∠H

GK

.

106.5

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

(con

tin

ued

)

Tra

pezo

ids a

nd

Kit

es

6-6

Exam

ple

1

Exam

ple

2

80°

60°

5 5

4

P

12

Lesson 6-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

39

G

len

co

e G

eo

me

try

ALG

EB

RA

F

ind

ea

ch

me

asu

re

.

1

. m

∠S

117

2

. m

∠M

38

63°

14

14

142°

21

21

3

. m

∠D

127

4

. R

H

√ #

#

544 =

4 √

##

34

36°

70°

20

12

12

ALG

EB

RA

Fo

r t

ra

pe

zo

id H

JK

L,

T a

nd

S a

re

mid

po

ints

of

the

le

gs.

5

. If

HJ

= 1

4 a

nd

LK

= 4

2,

fin

d T

S.

28

6

. If

LK

= 1

9 a

nd

TS

= 1

5,

fin

d H

J.

11

7

. If

HJ

= 7

an

d T

S =

10,

fin

d L

K.

13

8

. If

KL

= 1

7 a

nd

JH

= 9

, fi

nd

ST

.

13

CO

OR

DIN

AT

E G

EO

ME

TR

Y E

FG

H i

s a

qu

ad

ril

ate

ra

l w

ith

ve

rti

ce

s E

(1,

3),

F(5

, 0

),

G(8

, −

5),

H(−

4,

4).

9

. V

eri

fy t

hat

EF

GH

is

a t

rap

ezoid

.

−−

EF

‖ −

−

GH

, bu

t −

−

HE

∦ −

−

FG

10

. D

ete

rmin

e w

heth

er

EF

GH

is

an

iso

scele

s tr

ap

ezoid

. E

xp

lain

.

n

ot

iso

scele

s;

EH

= √

##

26 a

nd

FG

= √

##

34

6-6

Sk

ills

Pra

ctic

e

Tra

pezo

ids a

nd

Kit

es


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

40

G

len

co

e G

eo

me

try

Practi

ce

Tra

pezo

ids a

nd

Kit

es

Fin

d e

ach

me

asu

re

.

1

. m∠T

60

2

. m∠Y

112

60°

68°

7 7

3

. m∠Q

101

4

. BC

√

""

65

110°

48°

11

4

7 7

ALG

EB

RA

F

or t

ra

pe

zo

id FEDC

, V

an

d Y

are

mid

po

ints

of

the

le

gs.

5

. If

FE

= 1

8 a

nd

VY

= 2

8,

fin

d CD

. 38

6

. If

m∠F

= 1

40 a

nd

m∠E

= 1

25,

fin

d m∠D

. 55

CO

OR

DIN

AT

E G

EO

ME

TR

Y RSTU

is a

qu

ad

ril

ate

ra

l w

ith

ve

rti

ce

s R

(−3

, −

3),

S(5

, 1

),

T(1

0,

−2

), U

(−4

, −

9).

7

. V

eri

fy t

hat RSTU

is

a t

rap

ezoid

.

−−

RS

‖ −

−

TU

8

. D

ete

rmin

e w

heth

er RSTU

is

an

iso

scele

s tr

ap

ezoid

. E

xp

lain

.

n

ot

iso

scele

s;

RU

= √

""

37 a

nd

ST

= √

""

34

9

. C

ON

ST

RU

CT

ION

A

set

of

stair

s le

ad

ing t

o t

he e

ntr

an

ce o

f a b

uil

din

g i

s d

esi

gn

ed

in

th

e

shap

e o

f an

iso

scele

s tr

ap

ezoid

wit

h t

he l

on

ger

base

at

the b

ott

om

of

the s

tair

s an

d t

he

short

er

base

at

the t

op

. If

th

e b

ott

om

of

the s

tair

s is

21 f

eet

wid

e a

nd

th

e t

op

is

14 f

eet

wid

e,

fin

d t

he w

idth

of

the s

tair

s h

alf

way t

o t

he t

op

.

10

. D

ES

K T

OP

S A

carp

en

ter

need

s to

rep

lace

severa

l tr

ap

ezoid

-sh

ap

ed

desk

top

s in

a

class

room

. T

he c

arp

en

ter

kn

ow

s th

e l

en

gth

s of

both

base

s of

the d

esk

top

. W

hat

oth

er

measu

rem

en

ts,

if a

ny,

does

the c

arp

en

ter

need

?

6-6

FE

DC

VY

17.5

ft

Sam

ple

an

sw

er:

th

e m

easu

res o

f th

e b

ase a

ng

les

Lesson 6-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

41

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Tra

pezo

ids a

nd

Kit

es

1

. P

ER

SP

EC

TIV

E A

rtis

ts u

se d

iffe

ren

t

tech

niq

ues

to m

ak

e t

hin

gs

ap

pear

to b

e

3-d

imen

sion

al

wh

en

dra

win

g i

n t

wo

dim

en

sion

s. K

evin

dre

w t

he w

all

s of

a

room

. In

real

life

, all

of

the w

all

s are

rect

an

gle

s. I

n w

hat

shap

e d

id h

e d

raw

the s

ide w

all

s to

mak

e t

hem

ap

pear

3-d

imen

sion

al?

2.P

LA

ZA

In

ord

er

to g

ive t

he f

eeli

ng o

f

spaci

ou

sness

, an

arc

hit

ect

deci

des

to

mak

e a

pla

za i

n t

he s

hap

e o

f a k

ite.

Th

ree o

f th

e f

ou

r co

rners

of

the p

laza

are

sh

ow

n o

n t

he c

oord

inate

pla

ne.

If

the f

ou

rth

corn

er

is i

n t

he f

irst

qu

ad

ran

t, w

hat

are

its

coord

inate

s?

y

x

y

x

(6,

1)

3

. A

IRP

OR

TS

A

sim

pli

fied

dra

win

g o

f

the r

eef

run

way c

om

ple

x a

t H

on

olu

lu

Inte

rnati

on

al

Air

port

is

show

n b

elo

w.

How

man

y t

rap

ezoid

s are

th

ere

in

th

is

image?

4

. LIG

HT

ING

A

lig

ht

ou

tsid

e a

room

sh

ines

thro

ugh

th

e d

oor

an

d i

llu

min

ate

s a

trap

ezoid

al

regio

n ABCD

on

th

e f

loor.

AB

DC

Un

der

wh

at

circ

um

stan

ces

wou

ld

trap

ezoid

ABCD

be i

sosc

ele

s?

5. R

ISE

RS

A

ris

er

is d

esi

gn

ed

to e

levate

a s

peak

er.

Th

e r

iser

con

sist

s of

4 t

rap

ezoid

al

sect

ion

s th

at

can

be

stack

ed

on

e o

n t

op

of

the o

ther

to

pro

du

ce t

rap

ezoid

s of

vary

ing h

eig

hts

.

20 feet

10 feet

All

of

the s

tages

have t

he s

am

e h

eig

ht.

If a

ll f

ou

r st

ages

are

use

d,

the w

idth

of

the t

op

of

the r

iser

is 1

0 f

eet.

a.

If o

nly

th

e b

ott

om

tw

o s

tages

are

use

d,

wh

at

is t

he w

idth

of

the t

op

of

the r

esu

ltin

g r

iser?

b.

Wh

at

wou

ld b

e t

he w

idth

of

the r

iser

if t

he b

ott

om

th

ree s

tages

are

use

d?

6-6 tr

ap

ezo

ids

W

hen

th

e l

igh

t so

urc

e i

s a

n e

qu

al

dis

tan

ce f

rom

C a

nd

D,

sh

inin

g

str

aig

ht

thro

ug

h t

he d

oo

r.

15 f

t

12.5

ft5


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

42

G

len

co

e G

eo

me

try

En

ric

hm

en

t

Qu

ad

rila

tera

ls i

n C

on

str

ucti

on

Qu

ad

rila

tera

ls a

re o

ften

use

d i

n c

on

stru

ctio

n w

ork

.

1. T

he d

iagra

m a

t th

e r

igh

t re

pre

sen

ts a

roof

fr

am

e a

nd

sh

ow

s m

an

y q

uad

rila

tera

ls.

Fin

d

the f

oll

ow

ing s

hap

es

in t

he d

iagra

m a

nd

sh

ad

e i

n t

heir

ed

ges.

a.

isosc

ele

s tr

ian

gle

b.

scale

ne t

rian

gle

c.

rect

an

gle

d.

rhom

bu

s

e.

trap

ezoid

(n

ot

isosc

ele

s)

f.

isosc

ele

s tr

ap

ezoid

2. T

he f

igu

re a

t th

e r

igh

t re

pre

sen

ts a

win

dow

. T

he

wood

en

part

betw

een

th

e p

an

es

of

gla

ss i

s 3 i

nch

es

wid

e.

Th

e f

ram

e a

rou

nd

th

e o

ute

r ed

ge i

s 9 i

nch

es

wid

e.

Th

e o

uts

ide m

easu

rem

en

ts o

f th

e f

ram

e a

re

60 i

nch

es

by 8

1 i

nch

es.

Th

e h

eig

ht

of

the t

op

an

d

bott

om

pan

es

is t

he s

am

e.

Th

e t

op

th

ree p

an

es

are

th

e s

am

e s

ize.

a.

How

wid

e i

s th

e b

ott

om

pan

e o

f gla

ss?

b.

How

wid

e i

s each

top

pan

e o

f gla

ss?

c.

How

hig

h i

s each

pan

e o

f gla

ss?

3. E

ach

ed

ge o

f th

is b

ox h

as

been

rein

forc

ed

w

ith

a p

iece

of

tap

e.

Th

e b

ox i

s 10 i

nch

es

hig

h,

20 i

nch

es

wid

e,

an

d 1

2 i

nch

es

deep

. W

hat

is t

he l

en

gth

of

the t

ap

e t

hat

has

been

use

d?

jack

raf

ters

hip

raf

ter

com

mon r

afte

rs

Ro

of

Fra

me

ridge

boar

dva

lley

raft

er

pla

te

60 in

.

81 in

.

9 in

.

3 in

.

3 in

.

9 in

.

9 in

.

10 in

.

12 in

.

20 in

.

6-6

See s

tud

en

ts’

wo

rk.

42 i

n.

12 i

n.

30 i

n.

168 i

n.


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Inc.

Answers


Chapter 6 Assessment Answer Key

Quiz 2 (Lesson 6-3)

Page 45

Quiz 1 (Lessons 6-1 and 6-2) Quiz 3 (Lessons 6-4 and 6-5) Mid-Chapter Test

Page 45 Page 46 Page 47

Quiz 4 (Lesson 6-6)

Page 46

1.

2.

3.

4.

5.

12,240

45

20

(-1, 10)

A

1.

2.

3.

4.

5.

No; none of the

tests for s are ful! lled.

true

false

true

28 cm

1.

2.

3.

4.

5.

B

65

115

true

rectangle, rhombus, square

1.

2.

3.

4.

5.

118

√ ## 50 = 5 √ # 2

5

15.5

Use the distance formula to show CF = DE.

1.

2.

3.

4.

5.

B

J

A

G

C

6.

7.

8.

9.

10.

50

42

yes; opp. sides are $ to each other

30, 150

No; slope −−

XY = - 3 −

5

and slope of −−

WZ = - 1 −

3 ,

so opposite sides are

not parallel.

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c.


Chapter 6 Assessment Answer KeyVocabulary Test Form 1

Page 48 Page 49 Page 50

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

isosceles trapezoid

parallelogram

trapezoid

square

rhombus

false, rectangle

false; rhombus

diagonals

median

Sample answer:

angles formed by the

base and one of the

legs of the trapezoid

the nonparallel sides of a trapezoid

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

H

B

H

D

F

B

F

D

F

C

12.

13.

14.

15.

16.

17.

18.

19.

20.

H

B

J

A

G

A

J

A

G

B:

x = 7, m∠WYZ = 41

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Answers


Chapter 6 Assessment Answer KeyForm 2A Form 2B

Page 51 Page 52 Page 53 Page 54

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

G

D

F

C

F

A

H

B

F

D

12.

13.

14.

15.

16.

17.

18.

19.

20.

J

C

H

A

G

D

H

D

G

B: 7 or -4

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

J

D

F

B

G

A

H

C

J

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

G

D

G

D

H

C

F

A

H

B: 22

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ht ©

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c.


Chapter 6 Assessment Answer KeyForm 2C

Page 55 Page 56

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

1080

19

40

18

8

122

(6, 4)

Yes; −−

AB and −−

CD are

|| and ".

No; the slopes are

9 −

4 ,

1 −

7 , 1, and

2 −

3 .

Thus, ABCD doesnot have || sides.

- 2 − 3

22

Yes; if the diagonals

of a # are ", then

the # is a rectangle.

67.5

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

(4, 0)

31

Yes; ABCD has only one pair of opposite sides ||, −−

BC and −−

AD .

6

26

true

true

true

false

true

true

false

x = 9, y = 2

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Chapter 6 Assessment Answer KeyForm 2D

Page 57 Page 58

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

360

38

90

50

3.6

117

( 1 − 2 , 3)

Yes; Both pairs of

opp. sides are ".

ABCD has two pairs of

|| sides, −−

AB || −−

CD

and −−

BD || −−

DA ; it is a #.

-4

8

One rt. ∠ means that the other % will be rt. %. If all 4 % are

rt. %, the & is a rectangle.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

72

(-3, 1)

16

Yes; ABCD has only

one pair of opp.

sides ||, −−

AB and −−

BD .

8

17

false

true

false

false

true

false

true

90

Copyrig

ht ©

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c.


Chapter 6 Assessment Answer KeyForm 3

Page 59 Page 60

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

7

30; 30, 47, 120,

179, 174, and 170

180

− x

65

8 or 32

Yes; the diagonals bisect each other.

slope of −−

CD = 2 −

3 ;

slope of −−

DA = -2.

28

72

8 √ # 2

4

9

13

14.

15.

16.

17.

18.

19.

20.

B:

No; two pairs of congruent consecutive

sides do not exist.

CD = √ ## 72 , DA = √ ## 65 ,

AB = √ ## 20 , BC = √ ## 17

Yes; −−

AB ⊥ −−

BC ,

−− BC ⊥

−− CD ,

−− CD ⊥

−− AD ,

so opp % are &, and

all % are rt. %.

ABCD has 2 pairs of

opp. sides &, −−

AB

& −−

CD and −−

BC & −−

DA ,

so ABCD is a '.

105

Opp. sides of a ' are &.

If both pairs of opp. sides of a

quad. are &, then the quad.

is a '.

27

54

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Answers


Chapter 6 Assessment Answer Key

Extended-Response Test, Page 61

Scoring Rubric

Score General Description Speci! c Criteria

4 Superior

A correct solution that

is supported by well-

developed, accurate

explanations

• Shows thorough understanding of the concepts of angles of

polygons, properties of parallelograms, rectangles, rhombi,

squares, and trapezoids.

• Uses appropriate strategies to solve problems.

• Computations are correct.

• Written explanations are exemplary.

• Graphs and " gures are accurate and appropriate.

• Goes beyond requirements of some or all problems.

3 Satisfactory

A generally correct solution,

but may contain minor # aws

in reasoning or computation

• Shows an understanding of the concepts of angles of polygons,

properties of parallelograms, rectangles, rhombi, squares, and

trapezoids.

• Uses appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are effective.

• Graphs and " gures are mostly accurate and appropriate.

• Satis" es all requirements of problems.

2 Nearly Satisfactory

A partially correct

interpretation and/or solution

to the problem

• Shows an understanding of most of the concepts of angles of

polygons, properties of parallelograms, rectangles, rhombi,


• May not use appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are satisfactory.

• Graphs and " gures are mostly accurate.

• Satis" es the requirements of most of the problems.

1 Nearly Unsatisfactory

A correct solution with no

supporting evidence or

explanation

• Final computation is correct.

• No written explanations or work shown to substantiate the " nal

computation.

• Graphs and " gures may be accurate but lack detail or explanation.

• Satis" es minimal requirements of some of the problems.

0 Unsatisfactory

An incorrect solution

indicating no mathematical

understanding of the concept

or task, or no solution is

given

• Shows little or no understanding of most of the concepts of angles

of polygons, properties of parallelograms, rectangles, rhombi,


• Does not use appropriate strategies to solve problems.

• Computations are incorrect.

• Written explanations are unsatisfactory.

• Graphs and " gures are inaccurate or inappropriate.

• Does not satisfy requirements of problems.

• No answer may be given.

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Chapter 6 Assessment Answer Key Extended-Response Test, Page 61

Sample Answers

1. a. Any type of convex polygon can be drawn as long as one is regular and one is not regular and both have the same number of sides.

90°

70°

60°

110°120°

90°

90°

90°

b. Check to be sure that the exterior angles have been properly drawn and accurately measured.

c. 4(90) = 360; 120 + 70 + 60 + 110 = 360; The sum of the exterior angles of each figure should be 360°. The sum of the exterior angles of both the regular convex polygon and the irregular convex polygon is 360°.

2. The student should draw a rectangle and join the midpoints of consecutive sides. The figure formed inside is a rhombus. Since all four small triangles can be proved to be congruent by SAS, the four sides of the interior quadrilateral are congruent by CPCTC, making it a rhombus.

3. The student should draw an isosceles trapezoid with one pair of opposite sides parallel and the other pair of opposite sides congruent, as in the figure below.

4. a. Possible properties:A square has four congruent sides and a rectangle may not.A square has perpendicular diagonals and a rectangle may not.The diagonals of a square bisect the angles and those in a rectangle may not.

b. Possible properties:A square has four right angles and a rhombus may not.The diagonals of a square are congruent and those of a rhombus may not be.

c. Possible properties:A rectangle has four right angles and a parallelogram may not.The diagonals of a rectangle are congruent and those of a parallelogram may not be.

In addition to the scoring rubric found on page A30, the following sample answers

may be used as guidance in evaluating open-ended assessment items.

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Chapter 6 Assessment Answer KeyStandardized Test Practice

Page 62 Page 63

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

7

14.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9 0 0

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

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Chapter 6 Assessment Answer KeyStandardized Test Practice (continued)

Page 64

15.

16.

17.

18.

19.

20.

21. a.

b.

c.

hexagon;concave;irregular

88.9 cm

−−

DL

x = 11, y = 2

Assume thatneither bell costmore than $45.

40 cm3

−−

ML

−−

JL

J

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Chapter 6 Assessment Answer KeyUnit 2 Test

Page 65 Page 66

11.

12.

13.

14.

15.

16.

17.

18.

19.

yes

7 − 2 < x < 31

− 2

9

m∠JHK = 52;

m∠HMK = 108,

and x = 8

No; opp. side are

not ||.

5

5

11

In a parallelogram,

opposite sides are

congruent. Using

the distance

formula, PQ = √ ## 20 ,

PS = √ ## 20 , QR = √ # 8 ,RS = √ # 8 .

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

right; obtuse;

acute

m∠1 = 25; m∠2 = 25; m∠3 = 130

−−

AF % −−

ST , −−

FP % −−

TX , −−

PA % −−

XS

yes

ASA Postulate

∠A % ∠C

111

a = 9;

m∠ZWT = 32

∠YWZ

Neither appliance cost

more than $603.

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