Alternate Exterior Angles

Alternate Exterior Angles Definition

Alternate exterior angles are created when three lines intersect. A line that crosses two or more other lines is called a transversal. Often, two of the lines will be parallel, setting up some interesting angles with the transversal.

1. Definition
2. Theorem
3. Converse
4. Examples

When a transversal crosses two other lines, it creates an exterior and interior for the parallel lines. Alternate exterior angles are created in the space outside the parallel lines on alternating sides; interior angles are created in the space inside the parallel lines.

Here are lines $TR$ and $IP$, which would definitely cross somewhere in the distance. We cut across $TR$ and $IP$ with transversal $SW$, and where the transversal crosses $TR$ and $IP$, we have Points $L$ and $O$. Yep, it's a $SLOW$ $TRIP$.

We can write each angle with three letters: $\angle SLT$, $\angle SLR$, and so on down to $\angle POW$.

We can also use numbers in these same vertices, so $\angle SLT$ is $1$, $\angle SLR$ is $2$, and so on, ending with $\angle POW$ as $8$.

The four exterior angles:

1. $\angle 1$
2. $\angle 2$
3. $\angle 7$
4. $\angle 8$

Alternate exterior angles are pairs that appear outside the crossed lines and on different lines:

• $\angle 1 and \angle 8$
• $\angle 2 and \angle 7$

Alternate Exterior Angles Theorem

When two lines are parallel, the transversal creates congruent alternate exterior angles. The theorem says:

Here we have a new pair of lines, parallel and crossed by a transversal. We are not bothering to identify our parallel lines and transversal! We are jumping ahead to numbered angles.

The exterior angles are these same four:

1. $\angle 1$
2. $\angle 2$
3. $\angle 7$
4. $\angle 8$

This time, we can use the Alternate Exterior Angles Theorem to state that the alternate exterior angles are congruent:

• $\angle 1 \cong \angle 8$
• $\angle 2 \cong \angle 7$

Converse of the Alternate Exterior Angles Theorem

The converse of the Alternate Exterior Angles Theorem is also true:

Alternate Exterior Angles Examples

Begin by identifying alternate exterior angles, a common geometry problem. Here is another set of lines crossed by a transversal, with numbered angles:

Which are exterior angles?

Which are alternate exterior angles?

To find exterior angles, look in the space above and below the crossed lines. To find alternate exterior angles, look at that outside space for each crossed line, on different sides of the transversal.

We hope you said $\angle 1$, $\angle 2$, $\angle 7$, and $\angle 8$ are the exterior angles. We also hope you identified $\angle 1$ and $\angle 8$ as a pair of alternate exterior angles and $\angle 2$ with $\angle 7$ as the other pair.

Another type of problem is to identify the other angle that pairs with a given alternate exterior angle. Here are more lines and a transversal, with numbered angles:

To find the partner of $\angle 2$, look on the other side of the transversal. Look outside the crossed lines. Look at a crossed line that does not contain $\angle 2$.

Did you say $\angle 7$? We hope so!

Example With Parallel Lines

Here is a drawing with parallel lines and a transversal.

If we know that $\angle 8$ measures $130°$, what is the measure of $\angle 1$?

$\angle 8$ is on the outside of the bottom parallel line, and to the right of the transversal. That means $\angle 1$ is its alternate exterior angle partner. The Alternate Exterior Angles Theorem tells us it is also $130°$!

Lesson Summary

Now that you have gone through this lesson carefully, you are able to recall that angles on opposite sides of a transversal and outside two lines are called alternate exterior angles. When lines crossed by the transversal are parallel, you can use the Alternate Exterior Angles Theorem to know the alternate exterior angles are congruent. You can now solve problems identifying and measuring alternate exterior angles.

Geometry is a bit like a magician's trick. It sets up some odd event, like parallel lines being crossed by a transversal, and then hopes you're astounded when that improbable event leads to something new, like alternate exterior angles. Even when the lines are not parallel, alternate exterior angles exist, and we don't need to pull a rabbit out of a hat to amaze you with them.

Next Lesson:

Alternate Interior Angles