## What Are Alternate Exterior Angles?

**When two lines (****parallel lines****) are cut by a transversal at two distinct lines, a pair of ****angles ****that are created on the alternate sides of a transversal and on the outer side (exterior side) of the two lines are called alternate exterior angles.**

**What do alternate exterior angles look like?**

The diagram on the left side shows the alternate exterior angles formed by two non-parallel lines cut by transversal. The diagram on the right shows the alternate exterior angles formed by two parallel lines cut by transversal.

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## Alternate Exterior Angles: Definition

**Definition of alternate exterior angles: **A pair of angles formed on the alternate sides (opposite sides) of a transversal and on the exterior side of two lines cut by transversal are known as alternate exterior angles.

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## Alternate Exterior Angles Theorem

The alternate exterior angle theorem states that when two parallel lines are cut by a transversal, then the pair of alternate exterior angles formed are congruent.

## Proof of Alternate Exterior Angles Theorem

**Given**: l || m and t is a transversal.

Alt tag: Parallel lines l and m cut by a transversal

**To prove:**

m∠1 = m∠8

m∠2 = m∠7

**Proof:**

Statement | Reasoning |
---|---|

m∠1 = m∠4 | Vertically opposite angles |

m∠4 = m∠8 | Corresponding angles formed by two parallel lines cut by a transversal are congruent. |

m∠1 = m∠8 | by Transitive property |

Hence, proved.

## Converse of Alternate Exterior Angles Theorem

The converse of alternate exterior angles theorem states that if the alternate exterior angles formed by two lines cut by a transversal are congruent, then the two lines are parallel to one another.

## Facts about Alternate Exterior Angles

- Alternate exterior angles theorem is used to prove that two lines are parallel to each other.
- If the lines are not parallel, the alternate exterior angles are not congruent and do not have any relationship.

## Conclusion

In this article, we learned about** **Alternate Exterior Angles. We discussed the definition, the alternate exterior angles theorem, converse, and the proof. Let’s solve a few examples and practice problems for better comprehension.

## Solved Examples on Alternate Exterior Angles

**Example 1: Find the value of x.**

**Solution: **

m∠EFH = 130^{o}

m∠ACB = x

Here, m∠EFH + m∠GFH = 180^{o} …angles in a linear pair

Thus, 130 + m∠GFH = 180^{o}

m∠GFH = 180^{o} – 130^{o}

m∠GFH = 50^{o}

Now, m∠GFH = x …Alternate Exterior Angles Theorem

Thus, x = 50^{o}

**Example 2: If two alternate exterior angles formed by two lines and a transversal are ****(2x + 10) ****and ****(4x – 20)****, then what should the value of x be such that two lines are parallel?**

**Solution: **

For two lines to be parallel, the alternate exterior angles should be equal to each other.

2x + 10 = 4x – 20

4x – 2x = 20 + 10

2x = 30

$x = \frac{30}{2}$

x = 15

Thus, the two lines will be parallel for x = 15.

**Example 3: Find the value of y if the two parallel lines are cut by a transversal. **

**Solution:**

The given angles (3y + 49) and (7y – 55) are alternate exterior angles.

By the alternate exterior angles theorem, we have

(3y + 49) = (7y – 55)

4y = 104

y=26

## Practice Problems on Alternate Exterior Angles

## Alternate Exterior Angles - Definition, Theorem, Examples, FAQs

### Alternate exterior angles formed by two parallel lines cut by a transversal are

Alternate exterior angles are congruent when the two lines cut by a transversal are parallel.

### Find m∠2.

Alternate exterior angles formed by two parallel lines cut by a transversal are congruent.

The $60^{\circ}$ angle and ∠2 are alternate exterior angles formed by two parallel lines m and n cut by a transversal l are congruent.

Thus, m∠2 $= 60^{\circ}$

### Alternate exterior angles lie on the ______of the transversal.

Alternate exterior angles lie on the opposite sides of the transversal.

### Alternate exterior angles lie on the _____ of the parallel sides.

Alternate exterior angles lie on the outer side of the parallel sides.

## Frequently Asked Questions about Alternate Exterior Angles

**What are alternate interior angles?**

Alternate interior angles are a pair of angles that are on opposite sides of a transversal line and inside the two intersected lines.

**Is the alternate exterior angles theorem applicable when more than two parallel lines cut by a transversal?**

Yes, we can extend the alternate exterior angles theorem to more than two parallel lines cut by a transversal.

**Are alternate angles supplementary?**

Alternate angles are not supplementary angles, but they will add up to 180 if the transversal is perpendicular to the parallel lines.

**Are alternate exterior angles congruent?**

Alternate exterior angles formed when two parallel lines are cut by a transversal are congruent.