1. Lesson 6.5
Angles of Polygons pp

2. Objectives:
1. To prove three important theorems about the angles of triangles.
2. To develop a formula for the sum of the interior angles of a convex polygon.
3. To develop a formula for the measure of each interior angle of a regular polygon.

3. A B C X Y Z
An included side is the side between two consecutive angles. An included angle is an angle between two consecutive sides.

4. Theorem 6.16
The sum of the measures of the angles of any triangle is 180°.

5. Theorem 6.17
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

6. X Y Z A B C

7. Theorem 6.18 The acute angles of a right triangle are complementary. M

8. To find the total measure of the angles of a polygon:
1. Subdivide the polygon into triangles.
2. Multiply the number of triangles by °.
5(180) = 900°

9. Number of Sides Number of Triangles
4 2
5 3
6 4
7 5
8 6
9 7
10 8
n n - 2

10. EXAMPLE What is the sum of the measures of the angles of a pentagon
EXAMPLE What is the sum of the measures of the angles of a pentagon? If the pentagon is regular, find the measure of each angle.
3(180) = 540°
= 108°
540° 5

11. Practice: Find the measure of x.
48°
52°

12. Practice: Find the measure of x.
135°
60°

13. Practice: Find the measure of x.

14. Homework pp

15. ►A. Exercises
1. Name the included angle between AD and DC. A B C D E

16. ►A. Exercises
2. Name the included angle between EA and EB. A B C D E

17. ►A. Exercises
3. Name the included side between DAE and AED. A B C D E

18. ►A. Exercises
4. Name the included side between BCD and CDA. A B C D E

19. Give the sum of the measures of the angles in each convex polygon.
►A. Exercises
Give the sum of the measures of the angles in each convex polygon.
5. hexagon
6-2 = 4 (no. of triangles)
4(180) = 720°

20. Give the sum of the measures of the angles in each convex polygon.
►A. Exercises
Give the sum of the measures of the angles in each convex polygon.
6. decagon
10-2 = 8 (no. of triangles)
8(180) = 1440°

21. Give the sum of the measures of the angles in each convex polygon.
►A. Exercises
Give the sum of the measures of the angles in each convex polygon.
7. n-gon
n-2 = no. of triangles
180(n-2)°

22. Give the sum of the measures of the angles in each convex polygon.
►A. Exercises
Give the sum of the measures of the angles in each convex polygon.
sided polygon
100-2 = 98 (triangles)
98(180) = 17640°

23. Give the measure of an interior angle of each regular polygon.
►A. Exercises
Give the measure of an interior angle of each regular polygon.
9. heptagon
7-2 = 5 (no. of triangles)
5(180) = 900°
900°÷ 7 ≈ 128.6°

24. Give the measure of an interior angle of each regular polygon.
►A. Exercises
Give the measure of an interior angle of each regular polygon.
10. octagon
8-2 = 6 (no. of triangles)
6(180) = 1080°
1080°÷ 8 = 135°

25. Give the measure of an interior angle of each regular polygon.
►A. Exercises
Give the measure of an interior angle of each regular polygon.
11. n-gon
n-2 = no. of triangles
180(n-2)°
180(n-2)° ÷ n
(n-2)
180 n

26. Give the measure of an interior angle of each regular polygon.
►A. Exercises
Give the measure of an interior angle of each regular polygon.
sided polygon
84-2 = 82 (triangles)
82(180) = 14760°
14760 ÷ 84 ≈ 175.7°

27. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
13. mABE = 78; mBAE = 62. Find mAEB. A B C D E

28. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
14. mDEC = 56. Find mAED. A B C D E

29. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
15. mAED = 102; mEDA = 49. Find mDAE. A B C D E

30. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
16. mBEC = 82. Find mAED. A B C D E

31. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
17. Find mDEC + mECD + mCDE. A B C D E

32. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
18. mAEB = 40; mADE = 34. Find mEAD. A B C D E

33. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
19. mADC = 66; mADE = 28;
mAEB = 45. Find mECD. A B C D E

34. Use the figure to find the indicated measures.
►B. Exercises
Use the figure to find the indicated measures.
20. Find mABC + mBCD mCDA + mDAB. A B C D E

35. ■ Cumulative Review
State each postulate.
26. Ruler Postulate

36. ■ Cumulative Review
State each postulate.
27. Protractor Postulate

37. ■ Cumulative Review
State each postulate.
28. Completeness Postulate

38. ■ Cumulative Review
State each postulate.
29. Continuity Postulate

39. ■ Cumulative Review
State each postulate.
30. Parallel Postulate