AMSCO'S Geometry. New York - Rye High School

Interior and Exterior Angles of Polygons 369In general, the number of triangles into which the diagonals from a vertexseparate a polygon of n sides is two less than the number of sides, or n 2. Thesum of the interior angles of the polygon is the sum of the interior angles of thetriangles formed, or 180(n 2). We have just proved the following theorem:Theorem 9.16The sum of the measures of the interior angles of a polygon of n sides is180(n 2)°.Exterior Angles of a PolygonAt any vertex of a polygon, an exterior angle forms a linear pair with the interiorangle. The interior angle and the exterior angle are supplementary.Therefore, the sum of their measures is 180°. If a polygon has n sides, the sum ofthe interior and exterior angles of the polygon is 180n. Therefore, in a polygonwith n sides:The measures of the exterior angles 180n the measures of the interior angles 180n 180(n 2) 180n 180n 360 360We have just proved the following theorem:Theorem 9.17The sum of the measures of the exterior angles of a polygon is 360°.DEFINITIONA regular polygon is a polygon that is both equilateral and equiangular.If a triangle is equilateral, then it is equiangular. For polygons that havemore than three sides, the polygon can be equiangular and not be equilateral, orcan be equilateral and not be equiangular.Equilateralbut notequiangularEquiangularbut notequilateralEquiangularbut notequilateralEquilateralbut notequiangular