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.

Find the Interior Angles Sum of a Polygon

B. Find the measure of each interior angle of parallelogram RSTU.

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.

Step 1 Find x.

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.

Find the Interior Angles Sum of a Polygon

Step 2 Use the value of x to find the measure of each angle.

Answer: mR = 55, mS = 125, mT = 55, mU = 125

mR = 5x= 5(11) or 55

mS = 11x + 4= 11(11) + 4 or 125

mT = 5x= 5(11) or 55

mU = 11x + 4= 11(11) + 4 or 125

A. 900

B. 1080

C. 1260

D. 1440

A. Find the sum of the measures of the interior angles of a convex octagon.

A. x = 7.8

B. x = 22.2

C. x = 15

D. x = 10

B. Find the value of x.

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.

Interior Angle Measure of Regular Polygon

Understand Look at the diagram of the situation. The measure of the angle of a

corner inbetween two walkways is the interiorangle of a regular pentagon.

Plan Use the Polygon Interior Angles SumTheorem to find the sum of themeasures of the angles. Since theangles of a regular polygon arecongruent, divide this sum by thenumber of angles to find the

measureof each interior angle.

Interior Angle Measure of Regular Polygon

Solve Find the sum of the interiorangle measures.

(n – 2) ● 180 = (5 – 2) ● 180 n = 5

= 3 ● 180 or 540Simplify.

Find the measure of one interiorangle.

Substitution

Divide.

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 themeasure of each interior angle. Thelast side drawn should connect with

thebeginning point of the first segmentdrawn.

A. 130°

B. 128.57°

C. 140°

D. 125.5°

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.

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 SumTheorem

(150)n = 180(n – 2) S = 150n

150n = 180n – 360 Distributive Property

0 = 30n – 360 Subtract 150n from eachside.

Find Number of Sides Given Interior Angle Measure

Answer: The polygon has 12 sides.

360 = 30n Add 360 to each side.

12 = n Divide each side by 30.

A. 12

B. 9

C. 11

D. 10

The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

Find Exterior Angle Measures of a Polygon

A. Find the value of x in the diagram.

Find Exterior Angle Measures of a Polygon

Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x.

Answer: x = 12

5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) +

(5x + 5) = 360

(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360

31x – 12 = 360

31x = 372

x = 12

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 AngleSum Theorem

n = 36 Divide each side by 10.

Answer: The measure of each exterior angle of aregular decagon is 36.

A. 10

B. 12

C. 14

D. 15

A. Find the value of x in the diagram.

A. 72

B. 60

C. 45

D. 90

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