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CCSS I Can Statements: I can classify polygons by the number of sides I can classify polygons by the sum of the measures of the interior angles

Vocabulary Diagonal A Segment that connects any two non-consecutive vertices

Concept 1

ShapeNumber of Sides Sum of interior angles Triangle3 180 ˚ Quadrilateral4 Pentagon5 Hexagon6 Heptagon7 Octagon8 Nonagon9 Decagon10 Hendecagon11 Dodecagon12 N-gonn(n-2)×180

Example 1A 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) ● 180n = 9 =7 ● 180 or 1260Simplify. Answer:The sum of the measures is 1260.

Example 1B 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 1Find x.

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

Example 1A A.900 B.1080 C.1260 D.1440 A. Find the sum of the measures of the interior angles of a convex octagon.

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

Example 2 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 Interior Angle Measure of Regular Polygon UnderstandLook at the diagram of the situation. The measure of the angle of a corner in between two walkways is the interior angle of a regular pentagon. PlanUse the Polygon Interior Angles Sum Theorem to find the sum of the measures of the angles. Since the angles of a regular polygon are congruent, divide this sum by the number of angles to find the measure of each interior angle.

Example 2 Interior Angle Measure of Regular Polygon SolveFind the sum of the interior angle measures. (n – 2) ● 180= (5 – 2) ● 180n = 5 = 3 ● 180 or 540Simplify. Find the measure of one interior angle. Substitution Divide.

Example 2 Interior Angle Measure of Regular Polygon Answer:The measure of one of the interior angles of the food court is 108. CheckTo 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 A.130° B ° 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.

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 – 360Distributive Property 0=30n – 360Subtract 150n from each side.

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

Example 3 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.

Concept 2

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

Example 4A 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) (–12) ]=360 31x – 12=360 31x=372 x=12

Example 4B 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=360Polygon Exterior Angle Sum Theorem n=36Divide each side by 10. Answer:The measure of each exterior angle of a regular decagon is 36.

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

Example 4B A.72 B.60 C.45 D.90 B. Find the measure of each exterior angle of a regular pentagon.