6.1.1 Properties and Attributes of Polygons (continued) Example 2 Classifying Polygons In a regular polygon, every side has the same length and every angle has the same measure. In a convex polygon, the diagonals do not include points exterior to the polygon. In a concave polygon, diagonals do include exterior points. Regular polygons are always convex.
Three figures are identified as regular polygons or irregular polygons, and as concave or convex. The hatch marks indicate congruent side lengths. In the first and third polygons all sides are the same length. In the first figure all the angles are congruent, so this is a regular polygon. In the third figure, the angles are not all congruent, so this is an irregular polygon. The first figure is convex, because its diagonals do not include points on the exterior. Figure three is concave because some diagonals include exterior points. The second figure is an irregular polygon, because one of the sides is a different length from the other two. It is a convex polygon because the diagonals do not include exterior points.
Example 3 Finding Interior Angle Measures and Sums in Polygons
According to the Polygon Angle Sum Theorem, the sum of the interior angle measures of a convex polygon with n sides is ( n − 2)180°. To see why the theorem is true, draw diagonals from one vertex in a polygon. This forms n − 2 triangles in a polygon with n sides. For example, in the septagon, there are 5 triangles formed. Each triangle contains 180° of angles. The total measure of the angles within the polygon is the sum of the angles contained in all the triangles, or 5 times 180°.
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