Objectives Define polygon, concave / convex polygon, and regular polygon Find the sum of the measures of interior angles of a polygon Find the sum of the measures of exterior angles of a polygon

Definition of polygon A polygon is a closed plane figure formed by 3 or more sides that are line segments; the segments only intersect at endpoints no adjacent sides are collinear Polygons are named using letters of consecutive vertices

Concave and Convex Polygons A convex polygon has no diagonal with points outside the polygon A concave polygon has at least one diagonal with points outside the polygon

Regular Polygon Definition An equilateral polygon has all sides congruent An equiangular polygon has all angles congruent A regular polygon is both equilateral and equiangular Note: A regular polygon is always convex

Sum of Interior Angles in Polygons Convex Polygon # of Sides # of Triangles from 1 Vertex Sum of Interior Angle Measures Triangle 3 1 1* 180 = 180 Quadrilateral 4 2 2* 180 = 360 Pentagon 5 3* 180 = 540 Hexagon 6 4* 180 = 720 Heptagon 7 5* 180 = 900 Octagon 8 6* 180 = 1080 n-gon n n – 2 (n – 2) * 180

Example: Sum of Interior Angles Find m∠ X Solution: The sum of the measures of the interior angles for a quadrilateral is (4 – 2) * 180 = 360 The marks in the illustration indicate that m∠X = m∠Y. So the sum of all four interior angles is m∠X + m∠X + 100 + 90 = 360 2 m∠X + 190 = 360 2 m∠X = 170 m∠X = 85

Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex is 360. m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360

Example: Exterior Angle Sum What is the measure of an interior angle of a regular octagon? Solution: 8 * exterior angle = 360 (Ext. Angle Sum) exterior angle = 45 interior angle = 180 – exterior angle interior angle = 180 – 45 = 135