Geometry/Chapter 12

Interior angles are the angles inside a polygon. To find the sum of the interior angles, use the following expression: $(n-2)\cdot 180^{\circ }$ where $n$ is the number of sides of the polygon.

Example

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

$(5-2)\cdot 180^{\circ }=3\cdot 180^{\circ }=540^{\circ }$ 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.

${\frac {540^{\circ }}{5}}=108^{\circ }$

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: $35^{\circ }+75^{\circ }=110^{\circ }$ and $180^{\circ }-110^{\circ }=70^{\circ }$ 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?

$40^{\circ }-15^{\circ }=25^{\circ }$ 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: ${\frac {360^{\circ }}{15}}=24^{\circ }$ so each exterior angle is 24. The interior angles must add to 180 so $180^{\circ }-24^{\circ }=156^{\circ }$ so each interior angle is 156 degrees.