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"The main difference between a cat and a lie is that a cat only has nine lives." Mark Twain Chapter 6 Definitions • • • • • • • • Parallelogram Rectangle Rhombus Square Trapezoid Diagonal Base Legs Objective: Apply the Angle Sum Theorem for polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve contextual problems. Interior Angles of Polygons 180 180 180+ 180= 360 180 180 180 180+ 180 + 180 = 540 180 180 180 180 180+ 180 + 180 + 180 = 720 180+180+ 180 + 180 + 180 = 900 Interior Angles Sum Theoreom Convex Polygon # of Sides # of Triangles Sum of Angle Measures Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 Hexagon 6 4 720 Heptagon 7 5 900 Octagon 8 6 1080 Polygon N (n-2) 180(n-2) If a convex polygon has n sides and S is the sum of the measures of its interior angles then S=180(n-2) Application – Regular Polygon • The benzene molecule C6H6 consists of six carbon atoms in a regular hexagonal pattern with a hydrogen atom attached to each carbon atom. Find the sum of the measures of the interior angles of the hexagon. • • • • • Convex Polygon S=180(n-2) S=180(6-2) S=180(4) S=720 Application – Regular Polygon • A mall is designed so the five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of the measures of the interior angles of the pentagon. • • • • • Convex Polygon S=180(n-2) S=180(5-2) S=180(3) S=540 Irregular Polygon C B 2x x A • • • • • • 2x x D Find the measures of the angles. 360=mA + mB +mC + mD 360 = x + 2x +2x + x 360 = 6x x = 60 mA & mD = 60, mB & mC = 180 Exterior Angle Sum Theorem 1 If a polygon is convex, then the sum of the measure of the exterior angles, one at each vertex is 360 2 2 3 4 3 1 1 Exterior Angles • Find the measures of an exterior angle and an interior angle of the convex regular octagon ABCDEFGH Interior Angles S=180(n-2) Individual Interior Angles S=180(6) S=1080/8 Exterior Angles S=1080 = 135 360/8 = 45 Note Exterior & Interior Angle are a Linear Pair (supplementary) 135+45 = 180 Summary • For Convex Polygons • Measure of Interior Angles = 180(n-2) • Measure of Exterior Angles = 360 Practice Assignment Block Page 394 12 - 32 even