1. WARM-UP Tuesday, February 24, 2015
Classify each polygon by number of sides, regular or irregular, and concave or convex. 1. 3. 2. 4.

2. TEKS Objectives G.2A, G.3B, G.5A, G.5B, G9B
Find and use the measures of interior and exterior angles of polygons.

3. 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.

5. 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°.

6. 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.

7. 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.

8. 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.

9. Example 3C Continued
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°

10. 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.

11. 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.

12. 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°.

13. An exterior angle is formed by one side of a polygon and the extension of a consecutive side.
Remember!

15. 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°.

16. 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.

17. 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.
measure of one ext.
The measure of each exterior angle of a regular dodecagon is 30°.

18. 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.

19. 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.

20. 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.

21. 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°