Math 1312 Section 2.5 Convex Polygons Definitions: A polygon is a closed plane figure bounded by straight line segments as sides. A convex polygon has two properties: a) Every angle measures between 0o and 180o . b) A line segment joining two points of a convex polygon remains inside or on the boundary of the polygon. A concave polygon : a) Always has at least one reflex (between 180o and 360o ) angle. b) A line segment joining two points of a concave polygon can contain points in the exterior of the polygon. Example 1 : Convex polygon Concave Polygon Not a polygon Special names for polygons with fixed numbers of sides Number of sides Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 21 21-gon Definition : A diagonal of a polygon is a line segment that joins two nonconsecutive vertices. Example 2 : B A C D Quadrilateral 4 sides 2 diagonals Theorem : The total number of diagonals D in a polygon of n sides is given by the n(n − )3 formula D = . 2 Example 3 : Find the number of diagonals for any decagon. Theorem : The sum of measures of the interior angles of a polygon with n sides is given by S = (n − )2 ⋅180 o . Example 4 : Find the sum of measures of the interior angles of a pentagon. Example 5 : Find the number of sides in a polygon whose sum of interior angles is 1980o . Definition : A regular polygon is a polygon that is both equilateral (all sides are congruent) and equiangular (all angles are congruent). Corollary : The measure I of each interior angle of a regular polygon of n sides is (n − )2 ⋅180 o I = . n Example 6 : Find the measure of each interior angle of a stop sign. Example 7 : Find the number of sides that a regular polygon has if the measure of each interior angle is 144 o . Corollary : The sum of the four interior angles of a quadrilateral is 360o . Example 8 : The angle sum of a square is 360o . Corollary : The measure E of each exterior angle of a regular polygon of n sides is 360 o E = . n Example 9 : Find the measure of each exterior angle of a regular nonagon. Example 10 : Name the regular polygon whose exterior angle measures 120o . Observation : An interior of a polygon angle and an adjacent exterior angle are supplementary. 1 2 Example 11 : If an interior angle of a regular polygon measures 165o , find a) the measure of an exterior angle b) the number of sides .