Geometry Section 11.1 Angle Measures in Polygons

For each figure, draw all the diagonals from one vertex and complete the table.

Theorem 11.1 The sum of the measures of the interior angles of a (convex) polygon with n sides is

Corollary to Theorem 11.1 The measure of each interior angle of a regular n-gon is

Example 1: Find the sum of measures of the interior angles of a dodecagon. Example 2: Find the measure of each interior angle of a regular 20-gon.

While the sum of the interior angles of a polygon changes as the number of sides changes, this is not the case with the sum of the exterior angles.

Theorem 11.2 The sum of the measures of the exterior angles of a (convex) polygon, one at each vertex, with n sides is

Here’s an example of why that is the case Here’s an example of why that is the case. Adding the five equations together, we get:

Corollary to Theorem 11.2 The measure of each exterior angle of a regular n-gon is

n 168o Number of sides Sum of interior angles Measure of each exterior angles exterior angle n   16 3600o 14.4o 168o