1. Lesson 6-1 Angles of Polygons
TARGETS
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.
Targets

2. Polygon interior angles sume
LESSON 6.1: Angles of Polygons
Polygon Interior Angles Sum
# of Sides
Angle Measure
Sum of Angles 3
mA=
mC=
mB= 4
mF=
mH=
mG=
mI= 5
mP=
mS=
mQ=
mT=
mR=
Polygon interior angles sume

3. Polygon interior angles sume
LESSON 6.1: Angles of Polygons
Polygon Interior Angles Sum
The sum of the interior angle
measures of an n-sided convex
polygon is 180(n - 2)
Polygon interior angles sume

4. Answer: The sum of the measures is 1260.
LESSON 6.1: Angles of Polygons
EXAMPLE 1
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

5. B. Find the measure of each interior angle of parallelogram RSTU.
LESSON 6.1: Angles of Polygons
EXAMPLE 1
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

6. 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

7. 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

8. Find the measure of each interior angle of a regular octagon.
LESSON 6.1: Angles of Polygons
EXAMPLE 2
Interior Angle Measure of Regular Polygon
Find the measure of each interior angle of a regular octagon.
Example 2

9. S = 180(n – 2) Interior Angle Sum Theorem (150)n = 180(n – 2) S = 150n
LESSON 6.1: Angles of Polygons
EXAMPLE 3
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

10. 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

11. Polygon Exterior Angles Sum
LESSON 6.1: Angles of Polygons
Polygon Exterior Angles Sum
The sum of the exterior angle
measures of a convex polygon, one
angle at each vertex, is 360
Polygon Ext Angles Sum

12. A. Find the value of x in the diagram.
LESSON 6.1: Angles of Polygons
EXAMPLE 4
Find Exterior Angle Measures of a Polygon
A. Find the value of x in the diagram.
Example 4A

13. 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

14. B. Find the measure of each exterior angle of a regular decagon.
LESSON 6.1: Angles of Polygons
EXAMPLE 4
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