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Transcript

Definition Math 2 Geometry Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein 2.5 Convex Polygons A polygon is a closed plane figure whose sides are line segments that intersect only at the endpoints. In this text only: convex polygons (interior angles measure from 0° to 180°) Examples? Polygon Examples Not Polygons Convex Polygons Concave Polygons Classify by number of sides Triangle (3) Quadrilateral (4) Pentagon (5) Classify by number of sides Hexagon (6) Heptagon (7) Octagon (8) 1 Classify by number of sides Informal Definition A diagonal of a polygon is a line segment that joins two nonconsecutive vertices. Nonagon (9) B Decagon (10) A C E D Theorem 2.5.1 The total number of diagonals D in a polygon of n sides is given by the formula: D= n (n − 3) 2 Theorem 2.5.2 The sum S of the measures of the interior angles of a polygon with n sides is given by: S = (n – 2) 180° • Use the formula to find the number of diagonals of: • a triangle • a quadrilateral • a pentagon D= n (n − 3) 2 Informal Definitions • Equiangular polygon: All angles are congruent. • Equilateral polygon: The measure of all sides are equal. 2 Definition A regular polygon is a polygon that is both equilateral and equiangular. Not Regular Polygons Notice it is possible for Polygons to be either equilateral or equiangular but not both. Notice: A square is a regular quadrilateral. Rectangle Rhombus Corollary 2.5.3 The measure I of each interior angle of a regular polygon of n sides is: I= Corollary 2.5.4 The sum of the four interior angles of a quadrilateral is 360° (n − 2) ⋅180 n Corollary 2.5.5 The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. Corollary 2.5.6 The measure E of each exterior angle of a regular polygon of n sides is: E= 360 n 3 Informal Definition A polygram is the star-shaped figure that results when the sides of certain polygons are extended. Pentagram Hexagram 4