Classify polygons based on their sides and angles.

Find and use the measures of interior and exterior angles of polygons.

Objectives

side of a polygonvertex of a polygondiagonalregular polygonconcaveconvex

Vocabulary

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.

A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.

Remember!

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

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 2C: Classifying Polygons

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.

By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.

Remember!

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 trianglesis (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°

(7 – 2)180°

900°

Polygon Sum Thm.

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

Simplify.

Example 3C: Finding Interior Angle Measures and Sums in Polygons

Find the measure of each interior angle of pentagon ABCDE.

(5 – 2)180° = 540° Polygon Sum Thm.

mA + mB + mC + mD + mE = 540°Polygon Sum Thm.

35c + 18c + 32c + 32c + 18c = 540 Substitute.

135c = 540 Combine like terms.

c = 4 Divide both sides by 135.

Find the measure of each interior angle of a regular decagon.

Step 1 Find the sum of the interior angle measures.

Step 2 Find the measure of one interior angle.

Check It Out! Example 3b

(n – 2)180°

(10 – 2)180° = 1440°

Polygon Sum Thm.

Substitute 10 for n and simplify.

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°.

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°.

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°.

Polygon Sum Thm.

measure of one ext. =

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.

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.

Lesson Quiz

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°

24°