7.1 Angles of Polygons7.2 Properties of Parallelograms7.3 Proving That a Quadrilateral Is a Parallelogram7.4 Properties of Special Parallelograms7.5 Properties of Trapezoids and Kites

7 Quadrilaterals and OtherPolygons

Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

Gazebo (p. 369)

Amusement Park Ride (p. 381)

Window (p. 399)

Diamond (p. 410)

Arrow (p. 377)

Wi d ( 399)

Amusement Park Ride (p 381)

Gazebo (p. 369)

Arrow (p. 377)

DiDiamondd ((p. 41410)0)

SEE the Big Idea

361

Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency Using Structure to Solve a Multi-Step Equation (A.5.A)

Example 1 Solve 3(2 + x) = 9 by interpreting the expression 2 + x as a single quantity.

3(2 + x) = 9 Write the equation.

3(2 + x) 3 = 9

3 Divide each side by 3.

2 + x = 3 Simplify. 2 2 Subtract 2 from each side.

x = 5 Simplify.

Solve the equation by interpreting the expression in parentheses as asingle quantity.

1. 4(7 x) = 16 2. 7(1 x) + 2 = 19 3. 3(x 5) + 8(x 5) = 22

Identifying Parallel and Perpendicular Lines (G.2.B)

Example 2 Determine which of the lines are parallel and which are perpendicular.

Find the slope of each line.

Line a: m = 3 (3) 4 (2)

= 3

Line b: m = 1 (4) 1 2

= 3

Line c: m = 2 (2) 3 4

= 4

Line d: m = 2 0 2 (4)

= 1 3

Because lines a and b have the same slope, lines a and b are parallel. Because 1 3 (3) = 1, lines a and d are perpendicular and lines b and d are perpendicular.

Determine which lines are parallel and which are perpendicular.

4.

x

y4

2

2

4

(3, 4)

(1, 0)

(3, 3)

(3, 0)

(0, 4)(3, 2)

(2, 2) (4, 2)

a

b

c

d 5.

x

y4

2

31

(3, 1)

(4, 4)(0, 3)

(2, 1)

(2, 4)

(3, 3)

(0, 1)(4, 2)

abc

d

6.

7. ABSTRACT REASONING Explain why interpreting an expression as a single quantity does not contradict the order of operations.

x

y

2

4

2

42

(2, 2)

(1, 1)(4, 2)

(2, 4)(2, 3)

(4, 3)

(4, 0)

(3, 2)

b c

d

a

x

y4

2

2

424

(3, 1)

(0, 3)

(4, 4)

(2, 3)(3, 2)

(2, 2)

(2, 2)

(4, 4) c

d

a

b

362 Chapter 7 Quadrilaterals and Other Polygons

Mathematical Mathematical ThinkingThinkingMapping Relationships

Mathematically profi cient students create and use representations to organize, record, and communicate mathematical ideas. (G.1.E)

Monitoring ProgressMonitoring ProgressUse the Venn diagram above to decide whether each statement is true or false. Explain your reasoning.

1. Some trapezoids are kites. 2. No kites are parallelograms.

3. All parallelograms are rectangles. 4. Some quadrilaterals are squares.

5. Example 1 lists three true statements based on the Venn diagram above. Write six more true statements based on the Venn diagram.

6. A cyclic quadrilateral is a quadrilateral that can be circumscribed by a circle sothat the circle touches each vertex. Redraw the Venn diagram so that it includes cyclic quadrilaterals.

Writing Statements about Quadrilaterals

Use the Venn diagram above to write three true statements about different types of quadrilaterals.

SOLUTIONHere are three true statements that can be made about the relationships shown in the Venndiagram.

All rhombuses are parallelograms. Some rhombuses are rectangles. No trapezoids are parallelograms.

Classifi cations of Quadrilaterals

quadrilaterals

parallelograms

rhombuses rectangles

trapezoids

kites

squares

Core Core ConceptConcept

Section 7.1 Angles of Polygons 363

Angles of Polygons7.1

Essential QuestionEssential Question What is the sum of the measures of the interior angles of a polygon? of the exterior angles of a polygon?

Interior Angle Measures of a Polygon

Work with a partner. Use dynamic geometry software.

a. Draw a quadrilateral and a pentagon. Find the sum of the measures of the interior angles of each polygon.

Sample

A C

D

E

I

H

GF

B

b. Draw other polygons and fi nd the sums of the measures of their interior angles. Record your results in the table below.

Number of sides, n 3 4 5 6 7 8 9

Sum of interior angle measures, S

c. Plot the data from your table in a coordinate plane. What pattern(s) do you notice?

d. Write a function that fi ts the data. Explain what the function represents.

Exterior Angle Measures of a Polygon

Work with a partner.

a. Draw a quadrilateral and a pentagon. Find the sum of 1

2

3

4

Exterior Angles

the measures of the exterior angles of each polygon.

b. Draw other polygons and fi nd the sums of the measures of their exterior angles. Record your results in the table at the left.

c. Plot the data from your table in a coordinate plane. What pattern(s) do you notice?

d. Write a function that fi ts the data. Explain what the function represents.

Communicate Your AnswerCommunicate Your Answer 3. What is the sum of the measures of the interior angles of a polygon?

of the exterior angles of a polygon?

MAKING MATHEMATICAL ARGUMENTS

To be profi cient in math, you need to reason inductively about data.

Number of sides, n

Sum of exterior angle measures, M

3

4

5

6

7

8

9

G.5.A

TEXAS ESSENTIAL KNOWLEDGE AND SKILLS

364 Chapter 7 Quadrilaterals and Other Polygons

What You Will LearnWhat You Will Learn Use the interior angle measures of polygons.

Use the exterior angle measures of polygons.

Using Interior Angle Measures of PolygonsIn a polygon, two vertices that are endpoints of the same side are called consecutive vertices. A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

As you can see, the diagonals from one vertex divide a polygon into triangles. Dividing a polygon with n sides into (n 2) triangles shows that the sum of the measures of the interior angles of a polygon is a multiple of 180.

7.1 Lesson

Finding the Sum of Angle Measures in a Polygon

Find the sum of the measures of the interior angles of the fi gure.

SOLUTIONThe fi gure is a convex octagon. It has 8 sides. Use the Polygon Interior Angles Theorem.

(n 2) 180 = (8 2) 180 Substitute 8 for n. = 6 180 Subtract.

= 1080 Multiply.

The sum of the measures of the interior angles of the fi gure is 1080.

Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com 1. The coin shown is in the shape of an 11-gon. Find

the sum of the measures of the interior angles.

diagonal, p. 364equilateral polygon, p. 365equiangular polygon, p. 365regular polygon, p. 365

Previouspolygonconvexinterior anglesexterior angles

Core VocabularyCore Vocabullarry

TheoremTheoremTheorem 7.1 Polygon Interior Angles TheoremThe sum of the measures of the interior angles of a convex n-gon is (n 2) 180.

m1 + m2 + . . . + mn = (n 2) 180Proof Ex. 42 (for pentagons), p. 369

REMEMBERA polygon is convex when no line that contains a side of the polygon contains a point in the interior of the polygon.

D

EA

B

C

diagonals

A and B are consecutive vertices.

Vertex B has two diagonals, BD and BE .

Polygon ABCDE

12

3

456

n = 6

Section 7.1 Angles of Polygons 365

Finding an Unknown Interior Angle Measure

Find the value of x in the diagram.

SOLUTIONThe polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation.

x + 108 + 121 + 59 = 360 Corollary to the Polygon Interior Angles Theorem

x + 288 = 360 Combine like terms.

x = 72 Subtract 288 from each side.

The value of x is 72.

Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com 2. The sum of the measures of the interior angles of a convex polygon is 1440.

Classify the polygon by the number of sides.

3. The measures of the interior angles of a quadrilateral are x, 3x, 5x, and 7x. Find the measures of all the interior angles.

In an equilateral polygon, all sides are congruent.

In an equiangular polygon, all angles in the interior of the polygon are congruent.

A regular polygon is a convex polygon that is both equilateral and equiangular.

CorollaryCorollaryCorollary 7.1 Corollary to the Polygon Interior Angles TheoremThe sum of the measures of the interior angles of a quadrilateral is 360.

Proof Ex. 43, p. 370

Finding the Number of Sides of a Polygon

The sum of the measures of the interior angles of a convex polygon is 900. Classify the polygon by the number of sides.

SOLUTIONUse the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to fi nd the number of sides.

(n 2) 180 = 900 Polygon Interior Angles Theorem n 2 = 5 Divide each side by 180.

n = 7 Add 2 to each side.

The polygon has 7 sides. It is a heptagon.

108 121

59x

366 Chapter 7 Quadrilaterals and Other Polygons

Finding Angle Mea