1. Splash Screen

2. Five-Minute Check (over Chapter 5) Then/Now New Vocabulary
Theorem 6.1: Polygon Interior Angles Sum
Example 1: Find the Interior Angles Sum of a Polygon
Example 2: Real-World Example: Interior Angle Measure of Regular Polygon
Example 3: Find Number of Sides Given Interior Angle Measure
Theorem 6.2: Polygon Exterior Angles Sum
Example 4: Find Exterior Angle Measures of a Polygon
Lesson Menu

3. State whether this sentence is always, sometimes, or never true
State whether this sentence is always, sometimes, or never true. The three altitudes of a triangle intersect at a point inside the triangle.
A. always
B. sometimes
C. never A B C
5-Minute Check 1

4. Find n and list the sides of ΔPQR in order from shortest to longest if mP = 12n – 15, mQ = 7n + 26, and mR = 8n – 47.
A. n = 8;
B. n = 8;
C. n = 6;
D. n = 6; A B C D
5-Minute Check 2

5. State the assumption you would make to start an indirect proof of the statement. If –2x ≥ 18, then x ≤ –9.
A. x is positive.
B. –2x < 18
C. x > –9
D. x < 9 A B C D
5-Minute Check 3

6. Find the range for the measure of the third side of a triangle given that the measures of two sides are 43 and 29.
A. n > 14
B. 14 < n < 72
C. 29 < n < 43
D. n > 0 A B C D
5-Minute Check 4

7. A B C D Write an inequality relating mABD and mCBD.
A. mABD < mCBD
B. mABD ≤ mCBD
C. mABD > mCBD
D. mABD = mCBD A B C D
5-Minute Check 5

8. Write an equation that you can use to find the measures of the angles of the triangle.
A. x – 5 + 3x = 180
B. x – 5 + 3x = 180
C. x – 5 + 3x = 69
D. x – 5 + 3x = 111 A B C D
5-Minute Check 6

9. You named and classified polygons. (Lesson 1–6)
Find and use the sum of the measures of the interior angles of a polygon.
Find and use the sum of the measures of the exterior angles of a polygon.
Then/Now

10. diagonal
Vocabulary

11. Concept 1

12. Answer: The sum of the measures is 1260.
Find the Interior Angles Sum of a Polygon
A. Find the sum of the measures of the interior angles of a convex nonagon.
A nonagon has nine sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures.
(n – 2) ● 180 = (9 – 2) ● 180 n = 9
= 7 ● 180 or 1260 Simplify.
Answer: The sum of the measures is 1260.
Example 1A

13. B. Find the measure of each interior angle of parallelogram RSTU.
Find the Interior Angles Sum of a Polygon
B. Find the measure of each interior angle of parallelogram RSTU.
Step 1 Find x.
Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon.
Example 1B

14. Sum of measures of interior angles
Find the Interior Angles Sum of a Polygon
Sum of measures of interior angles
Substitution
Combine like terms.
Subtract 8 from each side.
Divide each side by 32.
Example 1B

15. Step 2 Use the value of x to find the measure of each angle.
Find the Interior Angles Sum of a Polygon
Step 2 Use the value of x to find the measure of each angle.
m  R = 5x
= 5(11) or 55
m  S = 11x + 4
= 11(11) + 4 or 125
m  T = 5x
= 5(11) or 55
m  U = 11x + 4
= 11(11) + 4 or 125
Answer: mR = 55, mS = 125, mT = 55, mU = 125
Example 1B

16. A. Find the sum of the measures of the interior angles of a convex octagon.
B. 1080
C. 1260
D. 1440 A B C D
Example 1A

17. A B C D B. Find the value of x. A. x = 7.8 B. x = 22.2 C. x = 15
D. x = 10 A B C D
Example 1B

18. Interior Angle Measure of Regular Polygon
ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the measure of one of the interior angles of the pentagon.
Example 2

19. Solve Find the sum of the interior angle measures.
Interior Angle Measure of Regular Polygon
Solve Find the sum of the interior angle measures.
(n – 2) ● 180 = (5 – 2) ● 180 n = 5
= 3 ● 180 or 540 Simplify.
Find the measure of one interior angle.
Substitution
Divide.
Answer: The measure of one of the interior angles of the food court is 108.
Example 2

20. A pottery mold makes bowls that are in the shape of a regular heptagon
A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl.
A. 130°
B °
C. 140°
D ° A B C D
Example 2

21. S = 180(n – 2) Interior Angle Sum Theorem (150)n = 180(n – 2) S = 150n
Find Number of Sides Given Interior Angle Measure
The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon.
Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides.
S = 180(n – 2) Interior Angle Sum Theorem
(150)n = 180(n – 2) S = 150n
150n = 180n – 360 Distributive Property
0 = 30n – 360 Subtract 150n from each side.
Example 3

22. Answer: The polygon has 12 sides.
Find Number of Sides Given Interior Angle Measure
360 = 30n Add 360 to each side.
12 = n Divide each side by 30.
Answer: The polygon has 12 sides.
Example 3

23. The measure of an interior angle of a regular polygon is 144
The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.
A. 12
B. 9
C. 11
D. 10 A B C D
Example 3

24. Concept 2

25. A. Find the value of x in the diagram.
Find Exterior Angle Measures of a Polygon
A. Find the value of x in the diagram.
Example 4A

26. Find Exterior Angle Measures of a Polygon
Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x.
5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) + (5x + 5) = 360
(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) (–12) ] = 360
31x – 12 = 360
31x = 372
x = 12
Answer: x = 12
Example 4A

27. B. Find the measure of each exterior angle of a regular decagon.
Find Exterior Angle Measures of a Polygon
B. Find the measure of each exterior angle of a regular decagon.
A regular decagon has 10 congruent sides and 10 congruent angles. The exterior angles are also congruent, since angles supplementary to congruent angles are congruent. Let n = the measure of each exterior angle and write and solve an equation.
10n = 360 Polygon Exterior Angle Sum Theorem
n = 36 Divide each side by 10.
Answer: The measure of each exterior angle of a regular decagon is 36.
Example 4B

28. A B C D A. Find the value of x in the diagram. A. 10 B. 12 C. 14 D. 15
Example 4A

29. B. Find the measure of each exterior angle of a regular pentagon.
Example 4B

30. End of the Lesson