• 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.

Lesson 6-1 Angles of Polygons

TARGETS

LESSON 6.1: Angles of Polygons

Polygon Interior Angles Sum

# of Sides Angle Measure Sum of Angles

3mA= mC=

mB=

4mF= mH=

mG= mI=

5

mP= mS=

mQ= mT=

mR=

LESSON 6.1: Angles of Polygons

Polygon Interior Angles Sum

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The sum of the interior anglemeasures of an n-sided convexpolygon is 180(n - 2)

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 1

LESSON 6.1: Angles of Polygons

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.

LESSON 6.1: Angles of Polygons

EXAMPLE 1

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

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

Interior Angle Measure of Regular Polygon

LESSON 6.1: Angles of Polygons

EXAMPLE 2

Find the measure of each interior angle of a regular octagon.

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.

EXAMPLE 3

LESSON 6.1: Angles of Polygons

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.

LESSON 6.1: Angles of Polygons

Polygon Exterior Angles Sum

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The sum of the exterior anglemeasures of a convex polygon, oneangle at each vertex, is 360

Find Exterior Angle Measures of a Polygon

A. Find the value of x in the diagram.

LESSON 6.1: Angles of Polygons

EXAMPLE 4

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.

LESSON 6.1: Angles of Polygons

EXAMPLE 4