Another way to approach the question of the sum of the vertex angles of an n-gon is to t...

Another way to approach the question of the sum of the vertex angles of an n-gon is to think of beginning at one vertex and traveling around the perimeter of the polygon. When you come to the next vertex, you must make a turn of a certain number of degrees. If you extend the edge you just traveled, you will note that there are two angles formed with respect to the vertex, the side you extended, and the next side. One of these is the vertex angle, and the other we will call the exterior angle as shown next

The measures of the vertex angle and the exterior angle add to 180°. Since there are n vertices, the sum of the measures of the vertex angles and exterior angles combined is n(180°). As you travel around the perimeter of the polygon, you must turn 360°. Therefore, the sum of the measures of the vertex angles in an n-gon is found to be

Since the sum of the measures of the n exterior angles of an n-gon is 360°, we have the following result for any regular n-gon:

a. What is the measure of an exterior angle for a regular hexagon?

b. If the measure of each vertex angle in a regular polygon is 144°, how many sides does the polygon have?

c. How many sides does a regular polygon have if an exterior angle has a measure of 40°?