1. Objectives Classify polygons based on their sides and angles.
Find and use the measures of interior and exterior angles of polygons.

2. Vocabulary side of a polygon vertex of a polygon diagonal
regular polygon
concave
convex

3. Each segment that forms a polygon is a side of the polygon
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.

4. You can name a polygon by the number of its sides
You can name a polygon by the number of its sides. The table shows the names of some common polygons.

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

6. 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

7. 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

8. All the sides are congruent in an equilateral 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.

9. 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.

10. Example 2A: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, convex

11. Example 2C: Classifying Polygons
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
regular, convex

12. Check It Out! Example 2b
Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
irregular, concave

13. 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.

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

16. 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°.

17. 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.

18. 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.

19. 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.

20. 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°.

22. 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.
measure of one ext.  =
The measure of each exterior angle of a regular 20-gon is 18°.

23. 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.

24. 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°
24°