6-1: Angles of Polygons yes no Definition of a quadrilateral - a closed figure with four sides and four vertices. Are these quadrilaterals? Yes or no?

Name a quadrilateral Name in order of the vertices. Quadrilateral HOPE Quadrilateral OPEH Quadrilateral PEHO Quadrilateral EHOP Quadrilateral HEPO Quadrilateral POHE H O E P

BL and LU are consecutive sides Nonconsecutive- not next to Consecutive- next to BL and LU are consecutive sides Nonconsecutive- not next to BL and EU are nonconsecutive sides Diagonals - segment whose endpoints are nonconsecutive vertices. BU and EL are diagonals B L E U

Theorem 6.1: Interior Angle Sum If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n - 2). Convex Polygon Number of Sides Number of Triangles Sum of Angle Measures Triangle 3 1 1 x 180 = 180 Quadrilateral 4 2 2 x 180 = 360 Pentagon 5 3 x 180 = 540 Hexagon 6 4 x 180 = 720 Heptagon 7 5 x 180 = 900 Octagon 8 6 x 180 = 1080

Example The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. S = 180 (n – 2) S = 135n 135n = 180 (n – 2) 135n = 180n – 360 -45n = -360 n = -360/-45 = 8 There are 8 sides

Example Find the measure of M in quadrilateral KLMN if mK = 2x, mL = 2x, mM = 2x-20, and mN = 3x + 20. 2x + 2x + 2x - 20 + 3x + 20 = 360 9x = 360 x = 40 Am I done? No!!! mM = 2x-20 = 2(40)-20 = 80-20 = 60

Theorem 6.2: Exterior Angle Sum If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360. m1 + m2 + m3 + m4 + m5 = 360 1 5 4 3 2

Example Find the measures of an exterior angle and an interior angle of a convex dodecagon. Hint: Dodecagon has 12 sides. Exterior Angle measure = 360/12 = 30 Interior Angle measure = (180*10)/12 = 150

Try these on Page 321 #1-6 540 3 4 mT = mV = 46; mU = mW = 134 60, 120 20, 160

Homework #37 P. 321 7-29 odd, 39-41