Geometry/Chapter 12

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Interior angles are the angles inside a polygon. To find interior angles, use the following expression: where is the number of sides of the polygon.

Example[edit]

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

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.

In a regular pentagon, each angle is 108 degrees

Sum of the Interior Angles of a Triangle[edit]

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: and so the third angle must be 70 degrees.

Triangle Exterior Angle Theorem[edit]

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?

So the other remote angle is 25 degrees.

The Sum of Exterior Angles Theorem[edit]

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

Exercises[edit]

Example Problem If a regular polygon has 15 sides, what is the measure of each exterior angle?

Answer: so each exterior angle is 24. The interior angles must add to 180 so so each interior angle is 156 degrees.

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