What if you were presented with two angles that are on opposite sides of a transversal, but outside the lines? How would you describe these angles and what could you conclude about their measures?

### Alternate Exterior Angles

**Alternate Exterior Angles** are two angles that are on the exterior of \begin{align*}l\end{align*} and \begin{align*}m\end{align*}, but on opposite sides of the transversal. ** \begin{align*}\angle 1\end{align*}** and

**are alternate exterior angles.**

*\begin{align*}\angle 8\end{align*}*

**Alternate Exterior Angles Theorem:** If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

The proof of this theorem is very similar to that of the Alternate Interior Angles Theorem.

**Converse of the Alternate Exterior Angles Theorem:** If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel.

#### Recognizing Alternate Exterior Angles

Using the picture above, list all the pairs of alternate exterior angles.

Alternate Exterior Angles: \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 7\end{align*}, \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 8\end{align*}.

#### Measuring Angles

Find \begin{align*}m \angle 1\end{align*} and \begin{align*}m \angle 3\end{align*}.

\begin{align*}m \angle 1 = 47^\circ\end{align*} because they are vertical angles. Because the lines are parallel, \begin{align*}m \angle 3 = 47^\circ\end{align*} by the Corresponding Angles Theorem. Therefore, \begin{align*}m \angle 2 = 47^\circ\end{align*}.

\begin{align*}\angle 1\end{align*} and \begin{align*}\angle 3\end{align*} are alternate exterior angles.

#### Real-World Application

The map below shows three roads in Julio’s town.

Julio used a surveying tool to measure two angles at the intersections in this picture he drew (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue.

The labeled \begin{align*}130^\circ\end{align*} angle and \begin{align*}\angle a\end{align*} are alternate exterior angles. If \begin{align*}m \angle a = 130^\circ\end{align*}, then the lines are parallel. To find \begin{align*}m \angle a\end{align*}, use the other labeled angle which is \begin{align*}40^\circ\end{align*}, and its linear pair. Therefore, \begin{align*}\angle a + 40^\circ = 180^\circ\end{align*} and \begin{align*}\angle a = 140^\circ\end{align*}. \begin{align*}140^\circ \neq 130^\circ\end{align*}, so Franklin Way and Chavez Avenue are not parallel streets.

### Examples

#### Example 1

Find the measure of each angle and the value of \begin{align*}y\end{align*}.

The given angles are alternate exterior angles. Because the lines are parallel, we can set the expressions equal to each other to solve the problem.

\begin{align*}(3y+53)^\circ & = (7y-55)^\circ\\ 108^\circ & = 4y\\ 27^\circ & = y\end{align*}

If \begin{align*}y = 27^\circ\end{align*}, then each angle is \begin{align*}3(27^\circ) + 53^\circ\end{align*}, or \begin{align*}134^\circ\end{align*}.

#### Example 2

Give THREE examples of pairs of alternate exterior angles in the diagram below:

There are many examples of alternate exterior angles in the diagram. Here are some possible answers:

- \begin{align*} \angle 1 \end{align*} and \begin{align*} \angle 14\end{align*}

- \begin{align*} \angle 2 \end{align*} and \begin{align*} \angle 13\end{align*}

- \begin{align*} \angle 12 \end{align*} and \begin{align*} \angle 13\end{align*}

### Review

- Find the value of \begin{align*}x\end{align*} if \begin{align*}m\angle 1 = (4x + 35)^\circ, \ m\angle 8 = (7x - 40)^\circ\end{align*}:
- Are lines 1 and 2 parallel? Why or why not?

For 3-8, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?

- \begin{align*}m\angle 2 = (8x)^\circ\end{align*} and \begin{align*}m\angle 7 = (11x-36)^\circ\end{align*}
- \begin{align*}m\angle 1 = (3x+5)^\circ\end{align*} and \begin{align*}m\angle 8 = (4x-3)^\circ\end{align*}
- \begin{align*}m\angle 2 = (6x-4)^\circ\end{align*} and \begin{align*}m\angle 7 = (5x+10)^\circ\end{align*}
- \begin{align*}m\angle 1 = (2x-5)^\circ\end{align*} and \begin{align*}m\angle 8 = (x)^\circ\end{align*}
- \begin{align*}m\angle 2 = (3x+50)^\circ\end{align*} and \begin{align*}m\angle 7 = (10x+1)^\circ\end{align*}
- \begin{align*}m\angle 1 = (2x-12)^\circ\end{align*} and \begin{align*}m\angle 8 = (x+1)^\circ\end{align*}

For 9-12, determine whether the statement is true or false.

- Alternate exterior angles are always congruent.
- If alternate exterior angles are congruent then lines are parallel.
- Alternate exterior angles are on the interior of two lines.
- Alternate exterior angles are on opposite sides of the transversal.

For questions 13-15, use the picture below.

- What is the alternate exterior angle with \begin{align*}\angle 2\end{align*}?
- What is the alternate exterior angle with \begin{align*}\angle 7\end{align*}?
- Are the two lines parallel? Explain.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.5.