# Geometry/Chapter 12

Interior angles are the angles inside a polygon. To find interior angles, use the following expression: (n-2) * 180 where n= sides of the polygon.

## Example

What is the sum of all the degrees in a pentagon?

(5-2)*180=3*180=540 degrees there are 540 degrees in a pentagon.

In order to find how many degrees are in each side of a regular pentagon (regular meaning same length and angle for each side), take the sum of all the interior angles and divide it by how many sides there are.

540/5=108

In a regular pentagon, each angle is 108 degrees

## Sum of the Interior Angles of a Triangle

The sum of the interior angles of a triangle is 180 degrees.

Example Problem:

What is the third angle of a triangle, given that the other two angles are 35 degrees and 75 degrees? Answer: 35 + 75 = 110 and 180 - 110 = 70 so the third angle must be 70 degrees.

## Triangle Exterior Angle Theorem

The exterior angle of a triangle is equal in measure to the sum of the two remote (not adjacent) interior angles of the triangle.

Example Problem:

If the exterior angle of a triangle is 40 degrees and if one of the remote angles is 15 degrees, what is the measure of the other remote angle? 40-15=25 So the other remote angle is 25 degrees.

## The Sum of Exterior Angles Theorem

The sum of exterior angles of a convex polygon taken one at each vertex is 360 degrees.

## Exercises

Example Problem If a regular polygon has 15 sides, what is the measure of each exterior angle? Answer: 360/15 = 24 so each exterior angle is 24. The interior angles must add to 180 so 180 - 24 = 156 so each interior angle is 156 degrees.