1. Five-Minute Check (over Chapter 5) CCSS 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

2. 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
5-Minute Check 6

3. 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
5-Minute Check 6

4. Mathematical Practices 4 Model with mathematics.
Content Standards
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
Mathematical Practices
4 Model with mathematics.
3 Construct viable arguments and critique the reasoning of others.
CCSS

5. You named and classified polygons.
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

6. diagonal
Vocabulary

7. Concept 1

8. 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:
Example 1A

9. 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

10. 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

11. 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

12. 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:
Example 1B

13. 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

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

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

16. 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

17. 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.
Example 2

18. Interior Angle Measure of Regular Polygon
Answer: The measure of one of the interior angles of the food court is 108.
Check To verify that this measure is correct, use a ruler and a protractor to draw a regular pentagon using 108 as the measure of each interior angle. The last side drawn should connect with the beginning point of the first segment drawn.
Example 2

19. Concept 2

20. 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

21. 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:
Example 4A

22. 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

23. 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:
Example 4B

24. 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

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

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

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

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