### Alternate Exterior Angles

**Alternate exterior angles** are two angles that are on the exterior of \begin{align*}l\end{align*}

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

If \begin{align*}l || m\end{align*}

**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.

If then \begin{align*}l || m\end{align*}

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

### Examples

For Examples 1 and 2, use the following diagram:

#### Example 1

Give an example of a pair of alternate exterior angles.

\begin{align*} \angle 1 \end{align*}

#### Example 2

Give another example of a pair of alternate exterior angles.

\begin{align*} \angle 2 \end{align*}

#### Example 3

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

The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.

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

If \begin{align*}y = 27\end{align*}

#### Example 4

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 \begin{align*}130^\circ\end{align*}

\begin{align*}\angle a + 40^\circ & = 180^\circ && \text{by the Linear Pair Postulate}\\
\angle a & = 140^\circ\end{align*}

\begin{align*}140^\circ \neq 130^\circ\end{align*}

#### Example 5

Which lines are parallel if \begin{align*}\angle AFG \cong \angle IJM\end{align*}

These two angles are alternate exterior angles so if they are congruent it means that \begin{align*}\overleftrightarrow{CG} || \overleftrightarrow{HK}\end{align*}

### Review

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

For 3-6, what does the value of \begin{align*}x\end{align*}

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

For 7-10, 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.

### Review (Answers)

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

### Resources