Geo Ch 07 TOC - edl...interior angles of a triangle to determine the sum of the measures of the...

Geometry Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 228

Chapter 7 Family and Community Involvement (English) .......................................... 229

Family and Community Involvement (Spanish) ......................................... 230

Section 7.1 ................................................................................................... 231

Section 7.2 ................................................................................................... 236

Section 7.3 ................................................................................................... 241

Section 7.4 ................................................................................................... 246

Section 7.5 ................................................................................................... 251

Cumulative Review ..................................................................................... 256

Copyright © Big Ideas Learning, LLC Geometry All rights reserved. Resources by Chapter

229

Chapter

7 Quadrilaterals and Other Polygons

Name _________________________________________________________ Date __________

Dear Family,

In this chapter, your student will learn the properties of shapes with four sides, which are also called quadrilaterals. The prefix “quad-” means “four” and the term “lateral” means “sides.” So, a quadrilateral is a shape with four sides.

One property of quadrilaterals your student will learn about is the relationship between diagonals. Find a square object such as a tabletop. Encourage your student to use measuring tape to find the lengths of the two diagonals of the square object.

• What do you notice about the lengths of the diagonals?

Find other square objects such as a napkin or the jewel case for a CD and repeat the experiment.

• Do the lengths of the diagonals show a general pattern?

Find several rectangular objects and measure the diagonals.

• Do the diagonals of rectangles follow the same pattern as the diagonals for squares?

Try to find other objects that have a surface that is a four-sided shape that is not a square or a rectangle. Measure and compare the diagonals.

• Do the patterns you noticed for squares and rectangles appear to extend to other types of quadrilaterals?

Have fun and be creative!

diagonals

diagonals


Capítulo

7 Cuadriláteros y otros polígonos

Nombre _______________________________________________________ Fecha ________

Estimada familia:

En este capítulo, su hijo aprenderá las propiedades de formas con cuatro lados, que también se llaman cuadriláteros. El prefijo “cuad-” significa “cuatro” y el término “látero” significa “lados”. Entonces, un cuadrilátero es una forma con cuatro lados.

Una propiedad de los cuadriláteros que aprenderá su hijo es la relación entre diagonales. Hallen un objeto cuadrado, tal como el tablero de una mesa. Haga que su hijo use cinta métrica para hallar las longitudes de las dos diagonales del objeto cuadrado.

• ¿Qué observan sobre las longitudes de las diagonales?

Hallen otros objetos cuadrados, tal como una servilleta o el estuche de un CD y repitan el experimento.

• ¿Las longitudes de las diagonales muestran un patrón en general?

Hallen varios objetos rectangulares y midan las diagonales.

• ¿Las diagonales de los rectángulos siguen el mismo patrón que las diagonales de los cuadrados?

Traten de hallar otros objetos que tengan una superficie que sea una forma con cuatro lados que no sea un cuadrado ni un rectángulo. Midan y comparen las diagonales.

• ¿Los patrones que observaron en los cuadrados y rectángulos parecen aplicarse a otros tipos de cuadriláteros?

¡Diviértanse y sean creativos!

diagonales

diagonales


231

7.1 Start Thinking

The polygon in the diagram has been formed by adjoining triangles. Use your knowledge of the sum of the measures of the interior angles of a triangle to determine the sum of the measures of the interior angles of the polygon.

1. Pentagon 2. Hexagon 3. Heptagon

Find the value of x in the diagram.

1. 2. 3.

Write an equation of the perpendicular bisector of the segment with endpoints P and Q.

1. ( ) ( )3, 2 , 5, 2P Q− − − 2. ( ) ( )5, 0 , 5, 2P Q −

3. ( ) ( )7, 4 , 3, 2P Q− 4. ( ) ( )8, 8 , 6, 3P Q−

7.1 Warm Up

7.1 Cumulative Review Warm Up

x°x°

70°64°

55° x°


7.1 Practice A

Name _________________________________________________________ Date _________

1. Find the sum of the measures of the interior angles of a heptagon.

2. The sum of the measures of the interior angles of a convex polygon is 3060 .°Classify the polygon by the number of sides.

3. Find the measure of each interior and exterior angle of a regular 30-gon.

In Exercises 4 and 5, find the value of x.

4. 5.

In Exercises 6 and 7, find the measures of ∠X and ∠Y.

6. 7.


8. 9.

10. A pentagon has three angles that are congruent and two other angles that are supplementary to each other. Find the measure of each of the three congruent angles in the pentagon.

11. You are designing an amusement park ride with cars that will spin in a circle around a center axis, and the cars are located at the vertices of a regular polygon. The sum of the measures of the angles’ vertices is 6120 .° If each car holds a maximum of four people, what is the maximum number of people who can be on the ride at one time?

120°W

X

Y

Z103°149°

V

W

X

Y

Z

U

108°

144°

122°

x°

100°

110°

160°

105°

105°

115°x°

60°

64°

36°

48°

x°x°

x°73°

109°


233

7.1 Practice B

Name _________________________________________________________ Date __________


1. 2.

In Exercises 3 and 4, find the measures of ∠X and ∠Y.

3. 4.


5. 6.

7. Find the measure of each interior angle and each exterior angle of a regular 24-gon.

8. Each exterior angle of a regular polygon has a measure of 18°. Find the number of sides of the regular polygon.

9. A polygon has two pairs of complementary interior angles and three sets of supplementary interior angles. The sum of the remaining interior angles is 1440 .°How many sides does the polygon have? Explain.

10. The figure shows interior angle measures of the kite.

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

b. Find the value of x.

127°

40°

x° 150° 102°

84°2x°

x°

130°

V

W Z

X Y

88° 100°

V U

W Z

YX

100°

92°

86°70°

x°

x°

(x + 30)°

x°x°

x°

x°

x°

110°x°

110°x°


7.1 Enrichment and Extension

Name _________________________________________________________ Date _________

Angles of Polygons In Exercises 1–8, use the figure to find the measure of the angle.

1. A∠

2. B∠

3. C∠

4. D∠

5. E∠

6. F∠

7. G∠

8. H∠

9. In an equiangular polygon, the measure of each exterior angle is 25% of the measure of each interior angle. What is the name of the polygon?

10. A and B are regular polygons and A has two more sides than B. The measure of each interior angle of A is six degrees greater than the measure of each interior angle of B. How many sides does A have?

11. The pentagon at the right has been dissected into three triangles with angles labeled as shown. Use the three triangles to prove that the sum of the interior angles of any pentagon is always 540 .°

58°

50°

50°

61°

40°

88°

70°

75°

A

H

D

E

B

C

G

F

ZV

W

X

A

B

CE

F

DG

H I

Y


235

Puzzle Time

Name _________________________________________________________ Date __________

Why Did The Pioneers Cross The Country In Covered Wagons?

A B C D E F

G H I J

Complete each exercise. Find the answer in the answer column. Write the word under the answer in the box containing the exercise letter.

7.1

Complete the sentence.

A. In a polygon, two vertices that are endpoints of the same side are called ____________ vertices.

B. A(n) _________of a polygon is a segment that joins two nonconsecutive vertices.

C. The sum of the measures of the interior angles of a(n) _____________ n-gon is ( )2 180 .n − • °

D. The sum of the measures of the _____ angles of a quadrilateral is 360 .°

E. The sum of the measures of the _______ angles of a convex polygon, one angle at each vertex, is 360 .°

Find the correct answer to the question for the interior angles of the convex polygon.

F. Two angles of a triangle measure 54 and 17 .° ° Find the measure of the third angle.

G. Find the sum of the measures of the interior angles of a 14-gon.

H. The sum of four angles in a pentagon is 440 .° Find the missing angle measure.

I. The sum of three angles in a pentagon is 320 ,° and the other two angles are ( ) ( )30 and 70 .x x+ ° − ° Find x.

J. What regular polygon has each interior angle measuring 135 ?°

interior

TO

consecutive

THEY

90°

THE

non-convex

NOW

120°

WEATHER

convex

WANT

decagon

FIRST

130°

A

289°

FOR

2160°

YEARS

concurrent

WAS

corner

INDIANS

exterior

WAIT

midsegment

GOLD

2520°

A

diagonal

DIDN’T

acute

HORSE

109°

FORTY

octagon

TRAIN

100°

FOR


7.2 Start Thinking

A scout is working on a construction project that involves building a 10-foot by 12-foot storage shed. He lays out a footprint of the building on the site using tent stakes and string, as shown in the diagram. The scout is certain of the measure of each side but does not have the proper tools to determine if the angles in each corner are right angles. Can the conclusion be made that the sides are definitely parallel? Consider how the scout could determine if the corner angles are right angles, by just using a tape measure.

Write a two-column proof.

1. Given: , MN PO NO MP≅ ≅

Prove: PMN NOP≅

2. Given: , , AB CD AB BD

CD BD

≅ ⊥

⊥

Prove: AD BC≅

Solve the equation. Justify each step.

1. 2 8 5 4x x− = + 2. ( )1 3 8 2 32

x x+ = −

3. 11 9 75

x x− = −

7.2 Warm Up


12 ft

10 ft

10 ft

12 ft

M

N

P

O

B

A C

D


237

7.2 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–4, find the value of each variable in the parallelogram.

1. 2.

3. 4.

5. Find the coordinates of the intersection of the diagonals of the parallelogram with vertices ( ) ( ) ( ) ( )2, 1 , 1, 3 , 6, 3 , and 3, 1 .− − −

In Exercises 6 and 7, three vertices of parallelogram ABCD are given. Find the remaining vertex.

6. ( ) ( ) ( )2, 0 , 2, 2 , 2, 2A B D− − − 7. ( ) ( ) ( )1, 3 , 1, 2 , 1, 2A C D− − − −

8. The measure of one interior angle of a parallelogram is 30° more than two times the measure of another angle. Find the measure of each angle of the parallelogram.

9. Your friend claims that you can prove that two parallelograms are congruent by proving that they have two pairs of congruent opposite angles. Is your friend correct? Explain your reasoning.

10. Use the diagram to write a two-column proof.

Given: PQRS is a parallelogram.

Prove: PQT RST≅

38

17 x + 3

y − 2

2u°

124°

(v − 3)°

27

3s

19

t + 5

P S

RQ

T

111°

3b°

15

a + 5


7.2 Practice B

Name _________________________________________________________ Date _________

In Exercises 1–4, find the value of each variable in the parallelogram.

1. 2.

3. 4.

5. Find the coordinates of the intersection of the diagonals of the parallelogram with vertices ( ) ( ) ( ) ( )2, 4 , 4, 4 , 2, 12 , and 4, 4 .− − −

6. Three vertices of parallelogram ABCD are ( ) ( ) ( )1, 5 , 1, 1 , and 2, 2 .A B D Find the coordinates of the remaining vertex.

7. Use the diagram to write a two-column proof.

Given: CEHF is a parallelogram. D bisects CE and G bisects .FH

Prove: CDF HGE≅

8. State whether each statement is always, sometimes, or never true for a parallelogram. Explain your reasoning.

a. The opposite sides are congruent.

b. All four sides are congruent.

c. The diagonals are congruent.

d. The opposite angles are congruent.

e. The adjacent angles are congruent.

f. The adjacent angles are complementary.

43

124 4(4y − 1)

3x + 10

66°3v°

u°

3b°3a + 5

(b + 84)°5a − 9

C D E

F G H

3c + 7

4c − 8

d12

d − 823


239


Name _________________________________________________________ Date __________

Properties of Parallelograms and Diagonals The given coordinates represent three vertices of a parallelogram. Write the coordinates of each other point that could be the fourth vertex.

1. ( ) ( ) ( )5, 1 , 2, 1 , 2, 7A B C− − − − − 2. ( ) ( ) ( )2, 5 , 1, 2 , 5, 1A B C−

3. ( ) ( ) ( ), , 2, , 4, 3A a b B a b C a b+ + + 4. ( ) ( ) ( )2 2 2, , , , , A a b B a b C a b

A diagonal is a line that connects one vertex of a polygon to a nonadjacent vertex. You can see from the picture below the diagonals drawn in a square, pentagon, and hexagon.

5. Complete the chart to the right to show the number of diagonals in each polygon.

6. Write a formula to find the number of diagonals in any n-gon.

7. How many diagonals does a decagon have? 13-gon?

8. If a polygon has 189 diagonals, how many sides does the polygon have?

9. There are six people in a tennis tournament who will play in round-robin, in which everyone has to play everyone else.

a. Draw a diagram that would represent this situation.

b. How many games will be played in this tennis tournament?

c. Write a simplified equation for the number of games played in round-robin play with n players.

Number of sides (n)

Number of diagonals (d)

3

4

5

6

7


Puzzle Time

Name _________________________________________________________ Date _________

Where Did Columbus Land When He Found America? Write the letter of each answer in the box containing the exercise number.


1. A ___________ is a quadrilateral with both pairs of opposite sides parallel.

2. If a _____________ is a parallelogram, then its opposite sides are congruent.

3. If a quadrilateral is a parallelogram, then its consecutive angles are _____________.

4. If a quadrilateral is a parallelogram, then its diagonals ____________ each other.

5. If a quadrilateral is a parallelogram, then its opposite angles are _____________.

Use the diagram.

6. 17, 14. Find .KG KJ GH= =

7. ( )86 , 6 . Find .m GKJ m GHJ x x∠ = ° ∠ = + °

8. 20. Find .KH KF=

9. 82 . Find .m HJK m GKJ∠ = ° ∠

10. 82 . Find .m HJK m HGK∠ = ° ∠

Answers

T. bisect A. 42°

D. 40 E. parallelogram

L. acute H. 82°

M. 17 O. supplementary

C. 98° D. intersect

E. 108° A. quadrilateral

H. congruent B. 14

T. 110° R. triangle

E. 10 C. polygon

S. complementary

N. 80

7.2

3 7 4 10 1 6 8 2 9 5

K J

F

G H


241

7.3 Start Thinking

Examine the diagram and determine if there appears to be enough information to conclude that the quadrilateral is a parallelogram. If there is not enough information, give an example of additional information that would allow you to prove the quadrilateral is a parallelogram.

1. 2. 3.

Use the points ( ) ( ) ( )A B C2, 5 , 5, 1 , 3, 2 ,− − and ( )D 1, 2− to find the indicated slope or measure.

1. Find the slope of .AB 2. Find the measure of .AC

3. Find the slope of .CD 4. Find the measure of .BD

5. Find the slope of .AC 6. Find the measure of .AB

For the conditional statement, write the converse, the inverse, and the contrapositive. Then determine if each statement is true.

1. If a triangle is right, then it contains two acute angles.

2. If two lines have the same slope, then they are parallel.

3. If there is ice on the road, then I will not go shopping.

7.3 Warm Up


A

D C

B

F G

E H

W Z

X Y


7.3 Practice A

Name _________________________________________________________ Date _________

In Exercises 1 and 2, state which theorem you can use to show that the quadrilateral is a parallelogram.

1. 2.

In Exercises 3 and 4, find the value of x that makes the quadrilateral a parallelogram.

3. 4.

In Exercises 5 and 6, graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram.

5. ( ) ( ) ( ) ( )4, 2 , 2, 1 , 4, 1 , 2, 2A B C D− − − − 6. ( ) ( ) ( ) ( )4, 1 , 1, 5 , 11, 0 , 8, 4E F G H− − −

7. Use the diagram to write a two-column proof. Given: A ABE∠ ≅ ∠

, AE CD BC DE≅ ≅

Prove: BCDE is a parallelogram.

8. In the diagram of the handrail for a staircase shown, 145 and .m A AB CD∠ = ° ≅

a. Explain how to show that ABDC is a parallelogram.

b. Describe how to prove that CDFE is a parallelogram.

c. Can you prove that EFHG is a parallelogram? Explain.

d. Find , , , and . m ACD m DCE m CEF m EFD∠ ∠ ∠ ∠

115°

115°

41

3x + 5

2x

2x

3x + 2 5x − 6

A

E D

B C

A

B

D

F

H

G

C

E


243

7.3 Practice B

Name _________________________________________________________ Date __________

In Exercises 1 and 2, state which theorem you can use to show that the quadrilateral is a parallelogram.

1. 2.

In Exercises 3 and 4, find the value of x that makes the quadrilateral a parallelogram.

3. 4.

In Exercises 5 and 6, graph the quadrilateral with the given vertices in a coordinate plane. Then show that the quadrilateral is a parallelogram.

5. ( ) ( ) ( ) ( )3, 1 , 3, 4 , 3, 2 , 3, 3W X Y Z− − − − 6. ( ) ( ) ( ) ( )4, 0 , 2, 2 , 5, 1 , 1, 3A B C D− − − −

7. Use the diagram to write a two-column proof. Given: A FDE∠ ≅ ∠

F is the midpoint of .AD

D is the midpoint of .CE

Prove: ABCD is a parallelogram.

8. A quadrilateral has two pairs of congruent angles. Can you determine whether the quadrilateral is a parallelogram? Explain your reasoning.

9. An octagon star is shown in the figure on the right.

a. Find , , and . m FCG m BCF m D∠ ∠ ∠

b. State which theorem you can use to show that the quadrilateral is a parallelogram.

c. The length of AB is three times the length of .AD Write an expression for the perimeter of parallelogram ABCD in terms of the variable x.

(2x + 5)°

3x°

5.8

5.8

7.27.2

7x + 1

8x − 10

D

F

A B

CE

45°45°

135°

E

C G

F

B

A D



Name _________________________________________________________ Date _________

Proving That a Quadrilateral Is a Parallelogram In Exercises 1–8, decide whether you are given enough information to determine that the quadrilateral is a parallelogram.

1. The opposite sides are parallel. 2. The opposite sides are congruent.

3. Two pairs of consecutive sides 4. Two pairs of consecutive angles are congruent. are congruent.

5. The diagonals are congruent. 6. The diagonals bisect each other.

7. All four sides are congruent. 8. The consecutive angles are supplementary.

9. If two opposite angles of a quadrilateral measure 120° and the measures of the other angles are multiples of 10, what is the probability that the quadrilateral is a parallelogram?

10. The diagonals of quadrilateral EFGH intersect at ( )1, 4 .D − Two vertices of EFGH are ( ) ( )2, 7 and 3, 5 .E F − What must be the coordinates of G and H to ensure that EFGH is a parallelogram?

11. In the diagram at the right, PQRS and QTSU are parallelograms. Is PTRU also a parallelogram? Explain why or why not.

12. Consider the supplementary angle relationships that you need to know to prove that a quadrilateral is a parallelogram. Make a conjecture using the least number of relationships that are necessary.

R

P

QX

S

T

U


245

Puzzle Time

Name _________________________________________________________ Date __________

What Kind Of Ship Can Last Forever? Circle the letter of each correct answer in the boxes below. The circled letters will spell out the answer to the riddle.


1. If both pairs of opposite sides of a quadrilateral are ___________, then the quadrilateral is a parallelogram.

2. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a __________.

3. If one _____ of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

4. If the diagonals of a quadrilateral ______ each other, then the quadrilateral is a parallelogram.

5. A quadrilateral is ________ a parallelogram.

Name the correct theorem number or give the correct value that would make the figure a parallelogram.

6. Given 72 ,m D∠ = ° find .m A∠

7. Given 89 , and ,m A m C m D m B∠ = ∠ = ° ∠ = ∠

indicate the theorem number that makes it a parallelogram.

8. 12, 12, 16. Find .DO BO AO CO= = =

9. 4 2, 5 3, . Find .DC x AB x AD CB x= + = − =

10. 2 1, 8, . Find .AD x CB x DC AB x= + = + =

7.3

F A R O R N I M S E

108° 7.7 always equal congruent side sometimes 12 72° parallelogram

I G N F D S H E I P

supplementary 6 pair intersect 16 7.8 bisect 24 7 5

A

B

OD

C


7.4 Start Thinking

A rhombus and a square are both quadrilaterals with four congruent sides, but a square always contains four right angles. Examine the diagrams below and determine some other distinctive characteristics of the rhombus and the square.

Use the diagrams to determine the measure of each angle.

1. 1m∠ 2. 2m∠ 3.

4. 5. 5m∠ 6. 6m∠

Determine whether the statement is always, sometimes, or never true. Explain your reasoning.

1. An isosceles triangle is a right triangle. 2. A right triangle is a scalene triangle. 3. An equilateral triangle is an equiangular triangle. 4. A right triangle is an equilateral triangle.

3m∠

4m∠

7.4 Warm Up


−5 5

−4

−2

4

2

y

x −2 2

−2

2

y

x

39°

2

3

1

22°

125°65

4


247

7.4 Practice A

Name _________________________________________________________ Date __________

In Exercises 1–5, the diagonals of rhombus ABCD intersect at E. Given that m EAD CE DE67 , 5, and 12,∠ = ° = = find the indicated measure.

1. m AED∠

2. m ADE∠

3. m BAE∠

4. AE

5. BE

In Exercises 6 and 7, find the lengths of the diagonals of rectangle JKLM.

6. 3 44 1

JL xKM x

= += −

7. 32

2 6

1

JL x

KM x

= −

= +

In Exercises 8 and 9, decide whether quadrilateral WXYZ is a rectangle, a rhombus, or a square. Give all names that apply. Explain your reasoning.

8. ( ) ( ) ( ) ( )3, 1 , 3, 2 , 5, 2 , 5, 1W X Y Z− − − − 9. ( ) ( ) ( ) ( )4, 1 , 1, 4 , 2, 1 , 1, 2W X Y Z− −

10. Use the figure to write a two-column proof. Given: PSUR is a rectangle.

PQ TU≅

Prove: QS RT≅

11. In the figure, all sides are congruent and all angles are right angles.

a. Determine whether the quadrilateral is a rectangle. Explain your reasoning.

b. Determine whether the quadrilateral is a rhombus. Explain your reasoning.

c. Determine whether the quadrilateral is a square. Explain your reasoning.

d. Find .m AEB∠

e. Find .m EAD∠

67°

12

5

E

A

CB

D

S

P Q R

T U

A

E

D

B C


7.4 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, decide whether quadrilateral JKLM is a rectangle, a rhombus, or a square. Give all names that apply. Explain your reasoning.

1. ( ) ( ) ( ) ( )3, 5 , 7, 6 , 6, 2 , 2, 1J K L M 2. ( ) ( ) ( ) ( )4, 1 , 1, 5 , 5, 2 , 2, 4J K L M− − − −

In Exercises 3–7, the diagonals of rhombus ABCD intersect at M. Given that m MAB MB AM53 , 16, and 12,∠ = ° = = find the indicated measure.

3. m AMD∠

4. m ADM∠

5. m ACD∠

6. DM

7. AC

8. Find the point of intersection of the diagonals of the rhombus with vertices ( ) ( ) ( ) ( )1, 2 , 3, 4 , 5, 8 , and 1, 6 .−

9. Use the figure to write a two-column proof. Given: WXYZ is a parallelogram. XWY XYW∠ ≅ ∠

Prove: WXYZ is a rhombus.

10. Your friend claims that you can transform every rhombus into a square using a similarity transformation. Is your friend correct? Explain your reasoning.

11. A quadrilateral has four congruent angles. Is the quadrilateral a parallelogram? Explain your reasoning.

12. A quadrilateral has two consecutive right angles. If the quadrilateral is not a rectangle, can it still be a parallelogram? Explain your reasoning.

13. Will a diagonal of a rectangle ever divide the rectangle into two isosceles triangles? Explain your reasoning.

X

W Z

Y

53°16

12

M

D C

BA


249


Name _________________________________________________________ Date __________

Properties of Special Parallelograms In Exercises 1–3, determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain.

1. The diagonals are congruent, but the quadrilateral has no right angles.

2. Each diagonal is 3 centimeters long and the two opposite sides are 2 centimeters long.

3. Two opposite angles are right angles, but the quadrilateral is not a rectangle.

In Exercises 4–7, use the information given in the diagram to solve for the missing variable.

4. Find the value of w.

5. Find the value of x.

6. Find the value of y.

7. Find the value of z.

8. In LMNP shown at the right, ( )232 , ,m MLN m NLP x∠ = ° ∠ = °

12 ,m MNP x∠ = ° and MNP∠ is an acute angle. Find .m NLP∠

9. Write a coordinate proof of part of Theorem 7.13 (Hint: write the vertices in terms of a and b.)

Given: DFGH is a parallelogram.

DG HF≅

Prove: DFGH is a rectangle.

L

P

M

Q

R

S25z

4x

31zy

9w − 1

7w + 17

N

−4

−2

4

−2 2

H(?, ?)

G(?, ?)

F(b, 0)

D(a, ?)y

L P

NM


Puzzle Time

Name _________________________________________________________ Date _________

What Do You Have To Know To Get Top Grades In Geometry? Write the letter of each answer in the box containing the exercise number.


1. A rhombus is a parallelogram with _______ congruent sides.

2. A rectangle is a parallelogram with four _______ angles.

3. A square is a _______ with four congruent sides and four right angles.

4. A parallelogram is a rhombus if and only if its _______ are perpendicular.

5. A parallelogram is a rhombus if and only if each diagonal _______ a pair of opposite angles.

6. A parallelogram is a rectangle if and only if its diagonals are _______ .

Decide whether each is a rhombus, rectangle, square, none of these, or all of these.

7. ( ) ( ) ( ) ( )8, 3 , 5, 3 , 1, 0 , 2, 6A B C D− − − − −

8. ( ) ( ) ( ) ( )6, 3 , 6, 8 , 2, 5 , 2, 0A B C D− − − − − − −

9. ( 7, 1), ( 4, 4), (2, 2), ( 3, 4)A B C D− − − −

Given rhombus ABCD, find the measure of the indicated angle in degrees.

10. 119 . Find .m A m B∠ = ° ∠

Find the length of the diagonals of rectangle QRST given the following information.

11. 4 6, 6 4QS x RT x= + = −

12. 9 12, 11 10QS x RT x= + = −

Answers

E. 61 H. diagonals

G. two A. four

A. square T. kite

L. parallelogram T. none

N. 111 O. acute

L. right D. intersects

G. rhombus R. rectangle

L. bisects M. all

E. 26 S. congruent

O. angles P. 119

I. 5 U. 11

R. perpendicular

7.4

7 2 5 9 4 11 1 12 8 3 10 6


251

7.5 Start Thinking

A kite is to be constructed according to the diagram with 1

41 yards of nylon fabric, one 38-inch dowel, and one 24-inch dowel. Describe the construction of the kite in geometric terms. Reference the segments and angles shown in the diagram.

Use the diagrams to determine the measure of the angle.

1. 1m∠ 2. 2m∠ 3. 3m∠

4. 4m∠ 5. 5m∠ 6. 6m∠

MN is a midsegment of ABC. Find the values of x and y.

1. 2. 3.

7.5 Warm Up


24 in.

38 in.

EA

B

D

C

60° 3

12

71°

45°

4

56

y

11

x

M

C

4

NA 6 B

N

M

A

y

x

10

12

BC

B

A

NMx

y

C34

9


7.5 Practice A

Name _________________________________________________________ Date _________


1. 2.

In Exercises 3 and 4, find the length of the midsegment of the trapezoid with the given vertices.

3. ( ) ( ) ( ) ( )0, 3 , 4, 5 , 4, 2 , 0, 2A B C D− − 4. ( ) ( ) ( ) ( )3, 3 , 1, 3 , 3, 3 , 5, 3E F G H− − − −

In Exercises 5 and 6, give the most specific name for the quadrilateral. Explain your reasoning.

5. 6.

7. Describe and correct the error in finding the most specfice name for the quadrialteral.

8. Use the diagram to write a two-column proof. Given: ABCD is a parallelogram.

AE AD≅

Prove: ABCE is an isosceles trapezoid.

9. The figure shows a window in the shape of a kite.

a. Find .m XVW∠

b. Find .XY

c. Which angle is congruent to ?XYZ∠

The quadrilateral has two pairs of consecutive congruent sides and the diagonals are perpendicular. So, the quadrilateral is a kite.

94°x° 100°

x°

W Z

X Y X Y

ZW

A B

E D C

X

V

Z

W Y

22 in.

26 in.


253

7.5 Practice B

Name _________________________________________________________ Date __________

In Exercises 1 and 2, show that the quadrilateral with the given vertices is a trapezoid. Then decide whether it is isosceles.

1. ( ) ( ) ( ) ( )1, 2 , 1, 3 , 3, 4 , 3, 3T U V W− − − − 2. ( ) ( ) ( ) ( )0, 0 , 2, 4 , 5, 4 , 5, 0P Q R S


3. 4.

In Exercises 5 and 6, give the most specific name for the quadrilateral. Explain your reasoning.

5. 6.

7. Use the diagram to write a two-column proof. Given: VXYZ is a kite.

, XY YZ WX UZ≅ ≅ Prove: WXV UZV≅

8. Three vertices of a trapezoid are given by ( ) ( ) ( )3, 6 , 3, 2 , and 6, 8 .− − − Find the fourth vertex such that the trapezoid is an isosceles trapezoid.

9. Is it possible to have a concave kite? Explain your reasoning.

10. The diagram shows isosceles trapezoid JKLP with base lengths a and b, and height c.

a. Explain how you know JKMN is a rectangle. Write the area of JKMN.

b. Write the formula for the area of .JNP

c. Write and simplify the formula for the area of trapezoid JKLP.

12 cm

X cm

22 cm

VW U

ZX

Y

P N M

KJ a

c

bL

55°

2x°

x°

B C

A D

D E

G F



Name _________________________________________________________ Date _________

Properties of Trapezoids and Kites 1. Each square section in an iron railing contains four

small kites. The figure shows the dimensions of one kite. What length of iron is needed to outline one small kite? How much iron is needed to outline one complete section, including the square?

2. Find the value of a in the figure to the right so that PQRS is isosceles.

3. The perimeter of an isosceles trapezoid ABCD is 27.4 inches. If ( )2 ,BC AB= find AD, AB, BC, and CD.

In Exercises 4 and 5, the given coordinates represent three vertices of an isosceles trapezoid. Write the coordinates of the point that could be the fourth vertex.

4. ( ) ( ) ( ), , , , 3, a b a b a b− + 5. ( ) ( ) ( ), , , , , 2a b a b c a c b c− − −

6. One base of a non-isosceles trapezoid has the vertices ( ), x y z+ and ( ), 2 .x z y z+ + A third vertex is the point ( ), .x y Describe the set of points that could be the fourth vertex.

7. If the coordinates ( ) ( ) ( )0, 0 , 2, 5 , and 5, 2 represent three vertices of a convex kite, describe the coordinates of each point that could be the fourth vertex.

7 in.

7 in.

7 in. 17 in.

P

S

Q

R

(a2 + 27)°

(2a2 − 54)°

B

EF

C

AD

8.62 in.


255

Puzzle Time

Name _________________________________________________________ Date __________

What Word Is Always Spelled Incorrectly? Circle the letter of each correct answer in the boxes below. The circled letters will spell out the answer to the riddle.


1. A ___________ is a quadrilateral with exactly one pair of parallel sides.

2. The parallel sides of a trapezoid are the ___________.

3. Base angles of a trapezoid are two ___________ angles whose common side is a base.

4. The nonparallel sides are the ___________ of the trapezoid.

5. If the legs of a trapezoid are congruent, then the trapezoid is an ___________ trapezoid.

6. A trapezoid is isosceles if and only if its ___________ are congruent.

7. The ___________ of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases.

8. If a quadrilateral is a ___________, then its diagonals are perpendicular.

Find the indicated measurement using quadrilateral ABCD as a reference.

9. , 75 . Find .AD BC m D m A≅ ∠ = ° ∠

10. 17, 25. Find .AB DC EF= =

Find the indicated measurement using quadrilateral ABCD as a reference.

11. , , 130 , 30 .

Find .

AD AB DC BC m A m Cm B≅ ≅ ∠ = ° ∠ = °

∠

7.5

G I N U C N O Y R R

9 kite 21 point trapezoid 18 100° 3 105° consecutive

E G C A T O L S T Y

bases 21 legs 1 midsegment median diagonals 0 altitude isosceles

D

A CE

B

CD

A B

FE


Chapter

7 Cumulative Review

Name _________________________________________________________ Date _________

In Exercises 1–16, solve the equation.

1. ( )6 2 18x − = 2. ( )4 7 36x− + =

3. ( )2 3 16x − = − 4. ( )4 6 24x− =

5. ( )9 8 63x− = − 6. ( )5 2 30x+ = −

7. ( )2 6 12 32x − + = 8. ( )8 2 12 4x + − = −

9. ( )3 10 16 20x − + = − 10. ( )6 8 11 1x− + = −

11. ( )2 9 13 15x− − − = − 12. ( )10 5 14 16x− − =

13. ( ) ( )4 11 2 9 44x x+ + + = 14. ( ) ( )3 10 8 1 11x x− + + =

15. ( ) ( )5 2 2 12 2x x− + + + = 16. ( ) ( )7 1 4 6 5x x− + + = −

In Exercises 17–22, classify the polygon.

17. 18. 19.

20. 21. 22.

23. The equation for the perimeter of a square can be expressed as ( )4 2 44.x + =

a. What is the value of x?

b. What is the side length of the square?

24. The length of a rectangle is 3 4x + and the width is 2 7.x +

a. Write an equation for the perimeter P of the rectangle.

b. The perimeter of the rectangle is 62 feet. What is the value of x?

c. What are the length and width of the rectangle?

6 6

6

10

4 4

10

7 7

7 7

8

3

8

3

8

6 10

5

5

5 5


257

Chapter

7 Cumulative Review (continued)

Name _________________________________________________________ Date __________

In Exercises 25–30, tell whether the figure to the right is a translation, reflection, rotation, or dilation of the figure to the left.

25. 26.

27. 28.

29. 30.

In Exercises 31–33, use the translation ( ) ( )x y x y, 2, 5 .→ + −

31. What is the image of ( )3, 7 ?A −

32. What is the image of ( )9, 8 ?B

33. What is the image of ( )4, 6 ?C −

In Exercises 34–37, use the translation ( ) ( )x y x y, 4, 3 .→ − +

34. What is the image of ( )8, 3 ?A −

35. What is the image of ( )12, 1 ?B − −

36. What is the preimage of ( )2, 8 ?C′ −

37. What is the preimage of ( )3, 8 ?D′


Chapter


Name _________________________________________________________ Date _________

In Exercises 38–41, find the measure of the exterior angle.

38. 39.

40. 41.

In Exercises 42 and 43, find the values of x and y.

42. EFG JKL≅ 43. VWX QRS≅

44. A right triangle has interior angles of ( )6 and 4 10 .x x° + °


b. What is the measure of the 6 angle?x°

c. What is the measure of the ( )4 10 angle?x + °

45. A right triangle has interior angles of ( ) ( )4 1 and 19 3 .x x+ ° − °


b. What is the measure of the ( )4 1 angle?x + °

c. What is the measure of the ( )19 3 angle?x − °

86° 2

53°

51°

(5x + 4)°

(10x − 5)°

(5x − 2)°

(14x − 7)°

(17x + 7)°

28°

62°

(7y − 1)°

K J

L

F

E

G

(5x + 13)°

18°7 feet

(4x − 2)°

W

V

X

S

R

Q(3y + 1) feet

77°

(7x + 7)°

2x°


259

Chapter


Name _________________________________________________________ Date __________

In Exercises 46–53, name the included angle between the pair of sides given.

46. GD and GF 47. EF and GF 48. GD and FD 49. FE and ED

50. ED and FD 51. DF and EF 52. ED and GD 53. GF and DF

In Exercises 54–57, find the indicated measure. Explain your reasoning.

54. SV 55. GH

56. KL 57. AD

58. A line segment AC is bisected by point B. The length of AB is 8 12x + and the length of BC is 6 18.x +


b. What is the length of ?AB

c. What is the length of ?AC

G F

ED

S T U

V

11.4

8.78.7

KJ

M

L

2x + 12

3x

G

E

HF

13.6

13.61.9

B CA

D

5x + 17 3x + 19

Geo Ch 07 TOC - edl...interior angles of a triangle to determine the sum of the measures of the...

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