# 4.2: Exterior Angles Theorems

**At Grade**Created by: CK-12

**Practice**Exterior Angles Theorems

What if you knew that two of the exterior angles of a triangle measured

### Watch This

CK-12 Foundation: Chapter4ExteriorAnglesTheoremsA

James Sousa: Introduction to the Exterior Angles of a Triangle

James Sousa: Proof that the Sum of the Exterior Angles of a Triangle is 360 Degrees

James Sousa: Proof of the Exterior Angles Theorem

### Guidance

An **exterior angle** is the angle formed by one side of a polygon and the extension of the adjacent side. In all polygons, there are **two** sets of exterior angles, one going around the polygon clockwise and the other goes around the polygon counterclockwise. By the definition, the interior angle and its adjacent exterior angle form a linear pair.

The **Exterior Angle Sum Theorem** states that each set of exterior angles of a polygon add up to

**Remote interior angles** are the two angles in a triangle that are not adjacent to the indicated exterior angle.

The **Exterior Angle Theorem** states that the sum of the remote interior angles is equal to the non-adjacent exterior angle. From the picture above, this means that

Given:

Prove:

Statement |
Reason |
---|---|

1. |
Given |

2. |
Triangle Sum Theorem |

3. |
Linear Pair Postulate |

4. |
Transitive PoE |

5. |
Subtraction PoE |

#### Example A

Find the measure of

If we draw both sets of exterior angles on the same triangle, we have the following figure:

Notice, at each vertex, the exterior angles are also vertical angles, therefore they are congruent.

#### Example B

Find the measure of the numbered interior and exterior angles in the triangle.

#### Example C

What is the value of

First, we need to find the missing exterior angle, we will call it

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter4TriangleSumTheoremA

#### Concept Problem Revisited

The third exterior angle of the triangle below is

By the Exterior Angle Sum Theorem:

### Vocabulary

** Interior angles** are the angles on the inside of a polygon while

**are the angles on the outside of a polygon.**

*exterior angles***are the two angles in a triangle that are not adjacent to the indicated exterior angle. Two angles that make a straight line form a**

*Remote interior angles***and thus add up to**

*linear pair***states that the three interior angles of any triangle will always add up to**

*Triangle Sum Theorem***states that each set of exterior angles of a polygon add up to**

*Exterior Angle Sum Theorem*### Guided Practice

1. Find

2. Find

3. Find the value of

**Answers:**

1. Set up an equation using the Exterior Angle Theorem.

2. Using the Exterior Angle Theorem,

3. Set up an equation using the Exterior Angle Theorem.

\begin{align*}&(4x+2)^\circ+(2x-9)^\circ =(5x+13)^\circ\\ & \quad \uparrow \qquad \nearrow \qquad \qquad \qquad \quad \uparrow\\ & \ \text{interior angles} \qquad \qquad \text{exterior angle}\\ & \qquad \qquad \quad (6x-7)^\circ = (5x+13)^\circ\\ &\qquad \qquad \qquad \qquad \ x = 20^\circ\end{align*}

Substituting \begin{align*}20^\circ\end{align*} back in for \begin{align*}x\end{align*}, the two interior angles are \begin{align*}(4(20)+2)^\circ =82^\circ\end{align*} and \begin{align*}(2(20)-9)^\circ=31^\circ\end{align*}. The exterior angle is \begin{align*}(5(20)+13)^\circ=113^\circ\end{align*}. Double-checking our work, notice that \begin{align*}82^\circ + 31^\circ = 113^\circ\end{align*}. If we had done the problem incorrectly, this check would not have worked.

### Interactive Practice

### Practice

Determine \begin{align*}m\angle{1}\end{align*}.

Use the following picture for the next three problems:

- What is \begin{align*}m\angle{1}+m\angle{2}+m\angle{3}\end{align*}?
- What is \begin{align*}m\angle{4}+m\angle{5}+m\angle{6}\end{align*}?
- What is \begin{align*}m\angle{7}+m\angle{8}+m\angle{9}\end{align*}?

Solve for \begin{align*}x\end{align*}.

- Suppose the measures of the three angles of a triangle are x, y, and z. Explain why \begin{align*}x+y+z=180\end{align*}.
- Suppose the measures of the three angles of a triangle are x, y, and z. Explain why the expression \begin{align*}(180-x)+(180-y)+(180-z)\end{align*} represents the sum of the exterior angles of the triangle.
- Use your answers to the previous two problems to help justify why the sum of the exterior angles of a triangle is 360 degrees. Hint: Use algebra to show that \begin{align*}(180-x)+(180-y)+(180-z)\end{align*} must equal 360 if \begin{align*}x+y+z=180\end{align*}.