Geometry/Chapter 12

Interior angles are the angles inside a polygon. To find the sum of the interior angles, use the following expression: $$(n-2)\cdot180^\circ$$ where $$n$$ is the number of sides of the polygon.

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

$$(5-2)\cdot180^\circ=3\cdot180^\circ=540^\circ$$ 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.

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?
 * Example Problem

$$40^\circ-15^\circ=25^\circ$$ 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.