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

2. 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.

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

4. Number of sides 3 4 5 6 7 8 9 10 11 12
Name of polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon

5. 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

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. 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

9. 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

10. 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

11. equilateral polygon
All the sides are congruent equiangular polygon
All the angles are congruent
regular polygon
Both equilateral and equiangular
*Note: If a polygon is not regular, it is called irregular.

12. convex - no diagonal contains points in the exterior
concave – any part of a diagonal contains points in the exterior of the polygon
convex - no diagonal contains points in the exterior
Note: A regular polygon is always convex.
Does that mean a irregular polygon is always concave?

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

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

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

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

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

18. Interior Angle Measures of
CONVEX POLYGONS
Draw all possible diagonals from one vertex of the polygon. This creates a set of triangles.

19. Interior Angle Measures of
CONVEX POLYGONS
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.

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

23. Example Find the sum of the interior angle measures of a
convex heptagon.
(n – 2)180°
(7 – 2)180°
900°

24. Example 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°
(16 – 2)180° = 2520°
Step 2 Find the measure of one interior angle.

25. Your Turn
Find the measure of each interior angle of a regular decagon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
(10 – 2)180° = 1440°
Step 2 Find the measure of one interior angle.

27. 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°.

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

29. Your Turn
Find the measure of each exterior angle of a regular dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360°.
measure of one ext.
The measure of each exterior angle of a regular dodecagon is 30°.

31. 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°