# 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.

## Contents

## Example[edit]

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[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: 35 + 75 = 110 and 180 - 110 = 70 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? 40-15=25 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: 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.

- Geometry Main Page
- Motivation
- Introduction
- Geometry/Chapter 1 Definitions and Reasoning (Introduction)
- Geometry/Chapter 1/Lesson 1 Introduction
- Geometry/Chapter 1/Lesson 2 Reasoning
- Geometry/Chapter 1/Lesson 3 Undefined Terms
- Geometry/Chapter 1/Lesson 4 Axioms/Postulates
- Geometry/Chapter 1/Lesson 5 Theorems
- Geometry/Chapter 1/Vocabulary Vocabulary

- Geometry/Chapter 2 Proofs
- Geometry/Chapter 3 Logical Arguments
- Geometry/Chapter 4 Congruence and Similarity
- Geometry/Chapter 5 Triangle: Congruence and Similiarity
- Geometry/Chapter 6 Triangle: Inequality Theorem
- Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
- Geometry/Chapter 8 Perimeters, Areas, Volumes
- Geometry/Chapter 9 Prisms, Pyramids, Spheres
- Geometry/Chapter 10 Polygons
- Geometry/Chapter 11
**Geometry/Chapter 12**Angles: Interior and Exterior- Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
- Geometry/Chapter 14 Pythagorean Theorem: Proof
- Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
- Geometry/Chapter 16 Constructions
- Geometry/Chapter 17 Coordinate Geometry
- Geometry/Chapter 18 Trigonometry
- Geometry/Chapter 19 Trigonometry: Solving Triangles
- Geometry/Chapter 20 Special Right Triangles
- Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
- Geometry/Chapter 22 Rigid Motion
- Geometry/Appendix A Formulas
- Geometry/Appendix B Answers to problems
- Appendix C. Geometry/Postulates & Definitions
- Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry