6-1: Angles of Polygons

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

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

3. 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

4. If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n - 2). Theorem 6.1: Interior Angle Sum

5. 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

6. 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

7. 1 5 2 4 3 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

8. 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

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

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