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## Sum of Interior and Exterior Angles in Polygons

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**Sum of Interior and Exterior Angles in Polygons**Essential Question – How can I find angle measures in polygons without using a protractor? Key Standard – MM1G3a**Polygons**• A polygon is a closed figure formed by a finite number of segments such that: 1. the sides that have a common endpoint are noncollinear, and 2. each side intersects exactly two other sides, but only at their endpoints.**Polygons**• Can be concave or convex. Concave Convex**Polygons are named by number of sides**Triangle 3 4 Quadrilateral Pentagon 5 Hexagon 6 Heptagon 7 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon**Regular Polygon**• A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular polygon.**Draw a:** Quadrilateral Pentagon Hexagon Heptagon Octogon • Then draw diagonals to create triangles. • A diagonal is a segment connecting two nonadjacent vertices (don’t let segments cross) • Add up the angles in all of the triangles in the figure to determine the sum of the angles in the polygon. • Complete this table**3**1 180° 4 2 2 · 180 = 360° 5 3 3 · 180 = 540° 4 4 · 180 = 720° 6 7 5 5 · 180 = 900° 8 6 6 · 180 = 1080° n n - 2 (n – 2) · 180°**Polygon Interior Angles Theorem**The sum of the measures of the interior angles of a convex n-gon is (n – 2) • 180. Examples – • Find the sum of the measures of the interior angles of a 16–gon. • If the sum of the measures of the interior angles of a convex polygon is 3600°, how many sides does the polygon have. • Solve for x. (16 – 2)*180 = 2520° (n – 2)*180 = 3600 180n = 3960 180 180 n = 22 sides 180n – 360 = 3600 + 360 + 360 (4 – 2)*180 = 360 4x - 2 108 108 + 82 + 4x – 2 + 2x + 10 = 360 6x = 162 6 6 2x + 10 82 6x + 198 = 360 x = 27**There are two sets of angles formed when the sides of a**polygon are extended. • The original angles are called interior angles. • The angles that are adjacent to the • interior angles are called exterior angles. • These exterior angles can be formed when any side is extended. • What do you notice about the interior angle and the exterior angle? • What is the measure of a line? • What is the sum of an interior angle with the exterior angle? Draw a quadrilateral and extend the sides. They form a line. 180° 180°**If you started at Point A, and followed along the sides of**the quadrilateral making the exterior turns that are marked, what would happen? You end up back where you started or you would make a circle. What is the measure of the degrees in a circle? A D B C 360°**Polygon Exterior Angles Theorem**• The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360°. • Each exterior angle of a regular polygon is 360 n where n is the number of sides in the polygon**Example**Find the value for x. Sum of exterior angles is 360° (4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360 7x + 248 = 360 – 248 – 248 7x = 112 7 7 x = 12 (4x – 12)⁰ 68⁰ 60⁰ 54⁰ (3x + 13)⁰ 65⁰ What is the sum of the exterior angles in an octagon? What is the measure of each exterior angle in a regular octagon? 360° 360°/8 = 45°**Classwork/Homework**Textbook: Read and study p298-299 Complete p300-301 (1-21) Show your work!