question - weebly

416 Chapter 8 Polygons and Area

Activity Angle Sum of PolygonsAngle Sum of PolygonsAngle Sum of Polygons8.28.2

Question

What is the sum of the measures of the interior angles of

a polygon with a given number of sides?

Think About It

1. Look for a pattern in the last column of the table. What is the sumof the measures of the interior angles of a heptagon (7-sidedpolygon)? of an octagon (8-sided polygon)? Put the answers inyour table.

2. Write an expression for the sum of the measures of the interiorangles of any convex polygon with n sides.

�3 Copy and complete the table below. Use the fact that the sum of the measures of the interior angles of a triangle is 180�.

• paper• pencil• straightedge

Materials

Polygon Number of Number of Sum of measuressides triangles of interior angles

Triangle 3 1 1 p 180� � 180�

Quadrilateral ? ? ?

Pentagon ? ? ?

Hexagon ? ? ?

Explore

�1 Using your straightedge, drawconvex polygons with threesides, four sides, five sides, andsix sides. A pentagon has beendrawn below.

�2 In each polygon, pick a vertexand draw the diagonals fromthat one vertex. Notice that thisdivides the pentagon into threetriangles.

cgpe-0802 08/16/2001 4:08 PM Page 416

8.2 Angles in Polygons 417

Goal

Find the measures ofinterior and exteriorangles of polygons.

Key Words

• interior angle p. 181

• exterior angle p. 181

The definitions for interior angles and exterior angles can be extendedto include angles formed in any polygon. In the diagrams shownbelow, interior angles are red, and exterior angles are blue.

exterior angles

interior angles

8.28.2 Angles in Polygons

Find the sum of the measures of theinterior angles of a convex heptagon.

Solution

A heptagon has 7 sides. Use the Polygon Interior Angles Theorem and substitute 7 for n.

(n � 2) p 180� � (7 � 2) p 180� Substitute 7 for n.

� 5 p 180� Simplify.

� 900� Multiply.

ANSWER � The sum of the measures of the interior angles of a convex heptagon is 900�.

EXAMPLE 111 Use the Polygon Interior Angles Theorem

Polygon Interior Angles Theorem

Words The sum of the measures of theinterior angles of a convex polygonwith n sides is (n � 2) p 180�.

Symbols ma1 � ma2 � . . . � man � (n � 2) p 180�

THEOREM 8.1

12 3

4

56 n � 6

Activity 8.2 suggests the Polygon Interior Angles Theorem shown below.

cgpe-0802 7/31/01 9:41 PM Page 417

In Exercises 1–3, find the measure of aG.

1. 2. 3.

4. Find the measure of an interior angle of a regular polygon withtwelve sides.

H

F

G

J

L M

K

140�

120�150�

140�

130�

H

G

J

K

103�133�

58�D H

GF

E

111�

118�

130� 109�


Find Measures of Interior Angles

Find the measure of aA in the diagram.

Solution

The polygon has 6 sides, so the sum of themeasures of the interior angles is:

(n � 2) p 180� � (6 � 2) p 180� � 4 p 180� � 720�.

Add the measures of the interior angles and set the sum equal to 720�.

136� � 136� � 88� � 142� � 105� � maA � 720� The sum is 720�.

607� � maA � 720� Simplify.

maA � 113� Subtract 607�.

ANSWER � The measure of aA is 113�.

EXAMPLE 222 Find the Measure of an Interior Angle

Find the measure of an interior angle ofa regular octagon.

Solution

The sum of the measures of the interiorangles of any octagon is:

(n � 2) p 180� � (8 � 2) p 180� � 6 p 180� � 1080�.

Because the octagon is regular, each angle has the same measure.So, divide 1080� by 8 to find the measure of one interior angle.

�10

880�� 135�

ANSWER � The measure of an interior angle of a regular octagon is 135�.

EXAMPLE 333 Interior Angles of a Regular Polygon

88� 142�

136�

136�

105�

D E

F

AB

C

SKILLS REVIEW

To review solving anequation, see p. 671.

Student Help

cgpe-0802 7/31/01 9:41 PM Page 418

Exterior Angles The diagrams below show that the sum of themeasures of the exterior angles of the convex polygon is 360�.

�1 Shade one exterior �2 Cut out the �3 Arrange the exteriorangle at each vertex. exterior angles. angles to form 360�.

The sum of the measures of the exterior angles of a convex polygon doesnot depend on the number of sides that the polygon has.

360�

1 5

4

3

2

1

4

5

2

3

1

4

5

2

3


Find the value of x.

Solution

Using the Polygon Exterior AnglesTheorem, set the sum of the measures of the exterior angles equal to 360�.

95� � 85� � 2x� � x� � 360� Polygon Exterior Angles Theorem

180 � 3x � 360 Combine like terms.

3x � 180 Subtract 180 from each side.

x � 60 Divide each side by 3.

ANSWER � The value of x is 60.

EXAMPLE 444 Find the Measure of an Exterior Angle

Polygon Exterior Angles Theorem

Words The sum of the measures of theexterior angles of a convex polygon,one angle at each vertex, is 360�.

Symbols ma1 � ma2 � . . . � man � 360�

THEOREM 8.2

1

23

4

5

85�

95�

x�

2x�

A circle contains twostraight angles. So,there are 180� � 180�,or 360�, in a circle.

Visualize It!

180�

180�

360�

or

n � 5

cgpe-0802 7/31/01 9:41 PM 19


1. Name an interior angle and an exterior angle of polygon ABCDEshown at the right.

2. Write the formula that is used to find the sum of the measures of the interior angles of any convex polygon with n sides.

3. Use your answer from Exercise 2 to find the sum of the measuresof the interior angles of a convex polygon with 9 sides.

Write an equation to find the measure of a1. Do not solve

the equation.

4. 5.

6. 7.

117�

57�59�

85�

186�115�

89�1

66�

130�

100�

140�65�

1

Skill Check

Vocabulary Check

Guided Practice

Exercises8.28.2

F DE H

G

C

B

A

Find the value of x.

5. 6.

7. 8.70�

65�2x� 75�

3x�

110�

x�x�

92�116�

3x �89�

46�

73�x�

60�

Find Exterior Angle Measures

cgpe-0802 7/31/01 9:41 PM Page 420


Extra Practice

See p. 689.

Example 1: Exs. 8–16Example 2: Exs. 17–20Example 3: Exs. 21–23Example 4: Exs. 24, 25

Homework Help

Interior Angles of Regular Polygons Find the measure of an interior

angle of the regular polygon.

21. 22. 23.

Using Algebra Find the value of x.

24. 25.

44�

85�

71�

3x�2x�

65�

76�

75� 44�

54� x�

Sum of Interior Angle Measures Find the sum of the measures of

the interior angles of the convex polygon.

8. 9. 10.

Polygons with n Sides Find the sum of the measures of the interior

angles of the convex polygon with n sides.

11. n � 10 12. n � 15 13. n � 20

14. n � 30 15. n � 52 16. n � 100

Interior Angle Measures Find the measure of aA.

17. 18.

19. 20.

158�

124�

120�146�102�

A

B C

D

E

F80� 130�

113�

A

B

CD

E

B

F

A

D

C

E135�

100�

91�

149�

110�B

A

CD101� 68�

92�

Practice and Applications

HOMEWORK HELP

Extra help with problemsolving in Exs. 21–23 is at classzone.com

IStudent HelpI C L A S S Z O N E . C O M

cgpe-0802 2/6/02 1:31 PM Page 421

34. Building Construction The metalframework for the windows shownincludes a regular hexagon. In thewindow frame, a1 c a2. Find themeasures of a1 and a2.

Challenge Find the number of sides of the regular polygon with the

given exterior angle measure.

35. 60� 36. 20� 37. 72� 38. 10�


Baseball A home plate for a baseball field is a pentagon as shown.

26. Is the polygon regular? Explain why or why not.

27. What is the sum of the interior angles?

28. Find the measures of aC and aE.

Cut a strip of lined paper and tie it into an overhand knot

as shown. Gently flatten the knot to form a pentagon.The pentagon

should be regular.

29. Find the measure of an interior angle of a regular pentagon.

30. Measure the interior angles of your knot with a protractor. Useyour answer to Exercise 29 to determine whether your pentagon is regular.

Logical Reasoning Select the word that makes the statement true.

31. The sum of the measures of the exterior angles of a convexpolygon, one at each vertex, is (always, sometimes, never) 360�.

32. The sum of the measures of the interior angles of a convexoctagon is (always, sometimes, never) 1440�.

33. The measures of the exterior angles of a convex polygon are(always, sometimes, never) equal.

Visualize It!

D

A

E

B

C

BASEBALL Home plate isused to find the placement ofthe foul lines on the playingfield.

Sports

21

LAHGE_10_CSse_cp.qxd 4/10/09 1:08 PM Page 422


Multiple Choice In Exercises 39 and 40, use the diagram below.

39. What is the value of x?

�A 18 �B 78

�C 117 �D 198

40. What is the measure of aP?

�F 39� �G 78� �H 156� �J 234�

Using a Centroid E is the centroid of TABC. Find BE and ED.

(Lesson 4.6)

41. BD � 6 42. BD � 33 43. BD � 52

Absolute Value Evaluate. (Skills Review, p. 662)

44. �5 45. 11 46. �48 47. 0

48. 13.2 49. �2 50. �0.001 51. �1.11

Decide whether the polygon is regular. Explain your answer.

(Lesson 8.1)

1. 2. 3.

Find the measure of a1. (Lesson 8.2)

4. 5. 6.

7. 8.

50�61�

44�

95� 193�

74�92�

1

104�

79�

68�

1100�

140� 120�

100�

140�1112�

76� 120�

92�

1

Quiz 1

Algebra Skills

CDA

B

E

A

B C

D

E

A

D

CBE

Mixed Review

Standardized Test

Practice R

S

TU

P

P

102�

2x �

130�

120�

134�

x �

cgpe-0802 7/31/01 9:42 PM Page 423

question - weebly

Documents