Alternate exterior angles are a pair of angles formed on the outside of two lines that are crossed by a third line. The alternate angles are located on opposite sides of the transverse line. Depending on the nature of the lines, the angles will have some characteristics. For example, if the two lines are parallel, the alternate exterior angles will be equal.

We will explore these types of angles in more detail below. We will also illustrate the concepts using diagrams and we will know the fundamental properties of these angles. In addition, we will learn about the alternate exterior angles theorem and we will look at some solved exercises.

## What are the alternate exterior angles?

Alternate exterior angles are angles created when three lines intersect. When a transverse line crosses the other two lines, it creates an exterior and an interior for the parallel lines. These angles are created in the space outside the parallel lines on alternate sides.

In the following diagram, we have two parallel lines that are crossed by a transverse line. We can list the outer vertices as follows:

Therefore, we have the four exterior angles:

- ∠1
- ∠2
- ∠3
- ∠4

Alternate exterior angles are pairs that appear outside the crossed lines and on different lines. In this case, the alternate exterior angles are:

- ∠1 and ∠4
- ∠2 and ∠3

## Properties of alternate exterior angles

The following are the fundamental properties of alternate exterior angles:

- When the two lines are parallel, the alternate exterior angles are congruent (they have the same measure).
- Consecutive exterior angles are supplementary (they add up to 180°).
- When the two lines are non-parallel, the alternate exterior angles have no specific properties.

## Alternate exterior angles theorem

When two lines are parallel, the transversal creates alternate exterior angles. The theorem says: “If a pair of parallel lines are crossed by a transversal, then the alternate exterior angles are congruent.”

In the following diagram, we have a pair of parallel lines that are crossed by a transversal.

We have the same exterior angles:

- ∠1
- ∠2
- ∠3
- ∠4

Using the alternate exterior angles theorem, we know that alternate exterior angles formed by parallel lines are congruent, so we have:

- ∠1 = ∠4
- ∠2 = ∠3

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## Inverse alternate exterior angles theorem

The opposite of the alternate exterior angles theorem is also true: “If the alternate exterior angles of two lines crossed by a transversal are congruent, then the lines are parallel.”

Therefore, if we have that the angles ∠1 and ∠4 are equal, we automatically know that the lines must be parallel.

## Alternate exterior angle examples with answers

We can use the properties of alternate exterior angles to solve some exercises.

### EXAMPLE 1

What is the value of X and Y in the following figure?

**Solution:** The angles 50° and Y are alternate exterior angles. Since the lines are parallel, we know that we have:

Y = 50°

Similarly, the angles 130° and Y are alternate exterior angles, so we have:

X = 130°

### EXAMPLE 2

What is the value of the missing angles in the diagram?

**Solution:** The angles ∠A, 110°, ∠C, and ∠D are exterior angles. Since the lines are parallel, we have:

∠C = 110°

Also, we know that consecutive angles are supplementary, so we have:

∠C + ∠D = 180°

∠D = 180° – ∠C = 180° – 110° = 70°

### EXAMPLE 3

We have that the angles (5*x*-25)° and (3x + 35)° are congruent alternate exterior angles. which is the value of *x*?

**Solution:** We know that congruent angles have the same measure, so we have:

5*x*-25 = 3*x*+35

5*x*-3*x* = 25+35

2*x* = 60

*x* = 30

Therefore, the angles given are:

5(30)-25 = 125°

### EXAMPLE 4

We have the consecutive exterior angles (3*x*-10)° and (*x*+30)°. Find the measure of the angles.

**Solution:** We know that consecutive exterior angles are supplementary, so we can form the following equation:

(3*x*-10) + (*x*+30) = 180

4*x*+20 = 180

4*x* = 160

*x* = 40

Therefore, the angles are:

3*x*-10 = 3(40)-10 = 110°

*x*+30 = 40+30 = 70°

## See also

Interested in learning more about angles? Take a look at these pages:

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