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6-1

Properties and Attributes of Polygons

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

Warm Up 1. A ? is a three-sided

polygon. 2. A ? is a four-sided

polygon. Evaluate each expression for n 6. 3.

(n 4) 12 4. (n 3) 90 Solve for a. 5. 12a

4a 9a 100

triangle

quadrilateral

24

270

4

Objectives

Classify polygons based on their sides and

angles. Find and use the measures of interior

and exterior angles of polygons.

Vocabulary

side of a polygon vertex of a polygon diagonal reg

ular polygon concave convex

In Lesson 2-4, you learned the definition of a

polygon. Now you will learn about the parts of a

polygon and about ways to classify polygons.

Each segment that forms a polygon is a side of

the polygon. The common endpoint of two sides is

a vertex of the polygon. A segment that connects

any two nonconsecutive vertices is a diagonal.

You can name a polygon by the number of its

sides. The table shows the names of some common

polygons.

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Example 1A Identifying Polygons

Tell whether the figure is a polygon. If it is a

polygon, name it by the number of sides.

polygon, hexagon

Example 1B Identifying Polygons

Tell whether the figure is a polygon. If it is a

polygon, name it by the number of sides.

polygon, heptagon

Example 1C Identifying Polygons

Tell whether the figure is a polygon. If it is a

polygon, name it by the number of sides.

not a polygon

Check It Out! Example 1a

Tell whether each figure is a polygon. If it is a

polygon, name it by the number of its sides.

not a polygon

Check It Out! Example 1b

Tell whether the figure is a polygon. If it is a

polygon, name it by the number of its sides.

polygon, nonagon

Check It Out! Example 1c

Tell whether the figure is a polygon. If it is a

polygon, name it by the number of its sides.

not a polygon

All the sides are congruent in an equilateral

polygon. All the angles are congruent in an

equiangular polygon. A regular polygon is one

that is both equilateral and equiangular. If a

polygon is not regular, it is called irregular.

A polygon is concave if any part of a diagonal

contains points in the exterior of the polygon.

If no diagonal contains points in the exterior,

then the polygon is convex. A regular polygon is

always convex.

Example 2A Classifying Polygons

Tell whether the polygon is regular or irregular.

Tell whether it is concave or convex.

irregular, convex

Example 2B Classifying Polygons

Tell whether the polygon is regular or irregular.

Tell whether it is concave or convex.

irregular, concave

Example 2C Classifying Polygons

Tell whether the polygon is regular or irregular.

Tell whether it is concave or convex.

regular, convex

Check It Out! Example 2a

Tell whether the polygon is regular or irregular.

Tell whether it is concave or convex.

regular, convex

Check It Out! Example 2b

Tell whether the polygon is regular or irregular.

Tell whether it is concave or convex.

irregular, concave

To find the sum of the interior angle measures of

a convex polygon, draw all possible diagonals

from one vertex of the polygon. This creates a

set of triangles. The sum of the angle measures

of all the triangles equals the sum of the angle

measures of the polygon.

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In each convex polygon, the number of triangles

formed is two less than the number of sides n. So

the sum of the angle measures of all these

triangles is (n 2)180.

Example 3A Finding Interior Angle Measures and

Sums in Polygons

Find the sum of the interior angle measures of a

convex heptagon.

(n 2)180

Polygon ? Sum Thm.

(7 2)180

A heptagon has 7 sides, so substitute 7 for n.

900

Simplify.

Example 3B Finding Interior Angle Measures and

Sums in Polygons

Find the measure of each interior angle of a

regular 16-gon.

Step 1 Find the sum of the interior angle

measures.

(n 2)180

Polygon ? Sum Thm.

Substitute 16 for n and simplify.

(16 2)180 2520

Step 2 Find the measure of one interior angle.

The int. ?s are ?, so divide by 16.

Example 3C Finding Interior Angle Measures and

Sums in Polygons

Find the measure of each interior angle of

pentagon ABCDE.

Polygon ? Sum Thm.

(5 2)180 540

Polygon ? Sum Thm.

m?A m?B m?C m?D m?E 540

35c 18c 32c 32c 18c 540

Substitute.

135c 540

Combine like terms.

c 4

Divide both sides by 135.

Example 3C Continued

m?A 35(4) 140

m?B m?E 18(4) 72

m?C m?D 32(4) 128

Check It Out! Example 3a

Find the sum of the interior angle measures of a

convex 15-gon.

(n 2)180

Polygon ? Sum Thm.

(15 2)180

A 15-gon has 15 sides, so substitute 15 for n.

2340

Simplify.

Check It Out! Example 3b

Find the measure of each interior angle of a

regular decagon.

Step 1 Find the sum of the interior angle

measures.

(n 2)180

Polygon ? Sum Thm.

Substitute 10 for n and simplify.

(10 2)180 1440

Step 2 Find the measure of one interior angle.

The int. ?s are ?, so divide by 10.

In the polygons below, an exterior angle has been

measured at each vertex. Notice that in each

case, the sum of the exterior angle measures is

360.

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Example 4A Finding Interior Angle Measures and

Sums in Polygons

Find the measure of each exterior angle of a

regular 20-gon.

A 20-gon has 20 sides and 20 vertices.

sum of ext. ?s 360.

Polygon ? Sum Thm.

A regular 20-gon has 20 ? ext. ?s, so divide the

sum by 20.

The measure of each exterior angle of a regular

20-gon is 18.

Example 4B Finding Interior Angle Measures and

Sums in Polygons

Find the value of b in polygon FGHJKL.

Polygon Ext. ? Sum Thm.

15b 18b 33b 16b 10b 28b 360

120b 360

Combine like terms.

b 3

Divide both sides by 120.

Check It Out! Example 4a

Find the measure of each exterior angle of a

regular dodecagon.

A dodecagon has 12 sides and 12 vertices.

sum of ext. ?s 360.

Polygon ? Sum Thm.

A regular dodecagon has 12 ? ext. ?s, so divide

the sum by 12.

The measure of each exterior angle of a regular

dodecagon is 30.

Check It Out! Example 4b

Find the value of r in polygon JKLM.

4r 7r 5r 8r 360

Polygon Ext. ? Sum Thm.

24r 360

Combine like terms.

r 15

Divide both sides by 24.

Example 5 Art Application

Ann is making paper stars for party decorations.

What is the measure of ?1?

?1 is an exterior angle of a regular pentagon. By

the Polygon Exterior Angle Sum Theorem, the sum

of the exterior angles measures is 360.

A regular pentagon has 5 ? ext. ?, so divide the

sum by 5.

Check It Out! Example 5

What if? Suppose the shutter were formed by 8

blades instead of 10 blades. What would the

measure of each exterior angle be?

?CBD is an exterior angle of a regular octagon.

By the Polygon Exterior Angle Sum Theorem, the

sum of the exterior angles measures is 360.

A regular octagon has 8 ? ext. ?, so divide the

sum by 8.

Lesson Quiz

1. Name the polygon by the number of its sides.

Then tell whether the polygon is regular or

irregular, concave or convex. 2. Find the sum of

the interior angle measures of a convex 11-gon.

nonagon irregular concave

1620

3. Find the measure of each interior angle of a

regular 18-gon. 4. Find the measure of each

exterior angle of a regular 15-gon.

160

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