# 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.

### Navigation

* Geometry Main Page
* Motivation
* Introduction
* Geometry/Chapter 1 - HS 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 $\mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}$
* 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 Formulae
* Geometry/Appendix B Answers to problems
* Appendix C. Geometry/Postulates & Definitions
* Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry
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