NAME DATE PERIOD 6-1 Angles of...

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Angles of Polygons6-1

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Chapter 6 93 Glencoe Geometry

What You’ll Learn Skim Lesson 6-1. Predict two things that you expect to learn based on the headings and figures in the lesson.

1. ____________________________________________________

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2. ____________________________________________________

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New Vocabulary Write the definition next to the term.

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Name all of the diagonals in polygon ABCDE.

Vocabulary Link Look up how television screens are measured. What does it mean for a television to have a 32-inch screen?

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diagonal

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093-104_GEOSNC06_890858.indd 93093-104_GEOSNC06_890858.indd 93 6/7/08 3:09:23 PM6/7/08 3:09:23 PM

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Lesson 6-1 (continued)

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Main Idea Details

Complete the following table for convex polygons. For middle column, find the number of triangles you can divide the polygon into by drawing all the possible diagonals from one vertex.

polygonnumberof sides

number of triangles

sum of interior angle measures

triangle

hexagon

octagon

nonagon

n-gon

Find the value of x in the figure below.

Polygon Interior Angles Sumpp. 389–392

Polygon Exterior Angles Sumpp. 392–393

(8x + 5)°(6x)°

80°

65°

Helping You Remember A good way to remember a new mathematical idea or formula is to relate it to something you already know. How can you use your knowledge of the Angle Sum Theorem (for a triangle) to help you remember the Interior Angle Sum Theorem?

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Parallelograms6-2

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What You’ll Learn Scan the text under the Now heading. List two things you will learn about in the lesson.

1. ____________________________________________________

____________________________________________________

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2. ____________________________________________________

____________________________________________________

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New Vocabulary Write the definition of the term.

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Fill in each blank using parallelogram ABCD.

−−

AB ||

−−− BE �

−−− AD �

−−− BC ||

−− AE �

parallelogram

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Main Idea Details

Complete the table using Theorems 6.3, 6.4, 6.5, and 6.6 in the Student Edition.

Properties of ParallelogramsTheorem Property

6.3 If a quadrilateral is a parallelogram, then its

opposite sides are .


opposite angles are .


opposite sides are .

6.6 If a parallelogram has one right angle, then it

has four .

Find the value of z in the parallelogram below.

Sides and Angles of Parallelogramspp. 399–400

Diagonals of Parallelogramspp. 401–402 9z - 4

4z + 21

Helping You Remember A good way to remember new theorems in geometry is to relate them to theorems you learned earlier. Name a theorem about parallel lines that can be used to remember the theorem that says, “If a parallelogram has one right angle, it has four right angles.”

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Tests for Parallelograms6-3

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What You’ll Learn Skim the Examples for Lesson 6-3. Predict two things you think you will learn about tests for parallelograms.

1. ____________________________________________________

____________________________________________________

____________________________________________________

2. ____________________________________________________

____________________________________________________

____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 6-1 and 6-2)

A diagonal is a segment in a polygon that connects any two

vertices.

A parallelogram is a quadrilateral with both pairs of

parallel.

Fill in each blank to review the properties of parallelograms.

The sides of a parallelogram are congruent.

The opposite angles of a parallelogram are .

Consecutive angles in a parallelogram are .

If a parallelogram has one angle, then it

has four angles.

diagonal

parallelogram

Active Vocabulary


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Main Idea Details

Complete the table using Theorems 6.9, 6.10, 6.11, and 6.12 in the student book for any quadrilateral ABCD.

Properties of Parallelograms

Theorem Property

6.9If both pairs of opposite sides of ABCD are

, then ABCD is a parallelogram.

6.10If both pairs of opposite angles of ABCD are


6.11If the diagonals of ABCD

each other, then ABCD is a parallelogram.

6.12If one pair of opposite sides of ABCD is both


Find the slope of each line segment to verify that WXYZ is a parallelogram.

Conditions for Parallelogramspp. 409–411

Parallelograms on the Coordinate Planepp. 412–413

y

x

slope of −−−

WX = _________

slope of −−

XY = _________

slope of −−

YZ = _________

slope of −−−

WZ = _________

Helping You Remember A good way to remember a large number of mathematical ideas is to think of them in groups. How can you state the conditions as one group about the sides of a quadrilateral that guarantee it is a parallelogram?

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Rectangles6-4

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What You’ll Learn Scan Lesson 6-4. List two headings you would use to make an outline of this lesson.

1. ____________________________________________________

____________________________________________________

2. ____________________________________________________

____________________________________________________

New Vocabulary Write the definition of the term.

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Fill in each blank using rectangle ABCD.

−− AB �� __________

−− AC � __________

m∠ADC = __________

m∠BCD = __________

−−−

AD � __________

rectangle

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Main Idea Details

Cross out the incorrect quadrilateral to complete the Venn diagram and illustrate the relationship between rectangles and parallelograms. Then write the definition of the correct quadrilateral in the space provided.

Use the Distance Formula to determine whether or not parallelogram RSTU is a rectangle.

Properties of Rectanglespp. 419–420

Prove That Parallelograms Are Rectanglespp. 420–421

rectangleparallelogram

rectangleparallelogram

y

xSU ≈ _________

RT ≈ _________

Helping You Remember It is easier to remember a large number of geometric relationships and theorems if you are able to combine some of them. How can you combine the two theorems about diagonals that you studied in this lesson?

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Rhombi and Squares6-5

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What You’ll Learn Scan the text in Lesson 6-5. Write two facts you learned about rhombi and squares.

1. ____________________________________________________

____________________________________________________

____________________________________________________

2. ____________________________________________________

____________________________________________________

____________________________________________________

New Vocabulary Label the diagrams with the correct terms.

Vocabulary Link A square is a shape that you learn at a very early age. Name some everyday items that are shaped like a square. Can you name any items that are shaped like a rhombus?

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rhombus

square

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Main Idea Details

Use the properties of rhombi to solve for x in rhombus ABCD.

Describe how you could prove that RS2 = RV2 + SV2 in rhombus RSTU.

Properties of Rhombi and Squarespp. 426–428

Prove That Quadrilaterals Are Rhombi or Squarespp. 428–430

AD = 8x - 11

DC = 5x + 13

Check

What property of rhombi can you use tohelp you solve for x?

Use the property to write an equation.

Solve for x.


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6-6

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Trapezoids and Kites6-6

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What You’ll Learn Skim the lesson. Write two things you already know about trapezoids and kites.

1. ____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

2. ____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

New Vocabulary Match the term with its definition by drawing a line to connect the two.

the segment that connects the midpoints of the legs of the trapezoid

a quadrilateral with exactly two pairs of consecutive congruent sides

the nonparallel sides of a trapezoid

the parallel sides of a trapezoid

the angles formed by the base and one of the legs of a trapezoid

a quadrilateral with exactly one pair of parallel sides

a trapezoid with congruent legs

base angles

bases

isosceles trapezoid

kite

legs of a trapezoid

midsegment of a trapezoid

trapezoid

Active Vocabulary


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Main Idea Details

Complete the flow proof below.Given: ABCD is an isosceles trapezoid

with bases −−

AB and −−−

CD .Prove: ∠BDC � ACD

Solve for x if −−−

MN is a midsegment of trapezoid RSTU.

Properties of Trapezoidspp. 435–438

Properties of Kitespp. 438–439

Given

Reflexive Prop. Diagonals are cong. Def. isoc. trap.

SSS CPCTC

x

16.5

14.4

x =

Helping You Remember A good way to remember a new geometric theorem is to relate it to one you already know. Name and state in words a theorem about triangles that is similar to the theorem in this lesson about the median of a trapezoid.

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NAME DATE PERIOD 6-1 Angles of...

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