# Alternate Interior Angles: in Parallel Lines

How to solve the alternate interior angles in parallel lines: formula, 2 examples, and their solutions.

## Alternate Interior Angles

### Definition

By two lines and a transversal,

two pairs of alternate interior angles are formed.

∠1 and ∠1'

∠2 and ∠2'

## Formula

### Formula

Alternate interior angles in parallel lines

are congruent.

m∠1 = m∠1'

m∠2 = m∠2'

## Example 1

### Example

### Solution

The given angles are

alternate interior angles in parallel lines.

So the given angles are congruent.

So [6x - 7] = [59].

Move -7 to the right side.

Then 6x = 66.

Divide both sides by 6.

Then x = 11.

So x = 11.

## Example 2

### Example

### Solution

Draw an auxiliary line

that is parallel to the horizontal lines

and that passes through the middle angle.

The upper two horizontal lines are parallel.

So these two blue angles are

alternate interior angles in parallel lines.

So the blue angles are congruent.

The lower two horizontal lines are parallel.

So these two green angles are

alternate interior angles in parallel lines.

So the green angles are congruent.

See the middle angle.

The middle angle is formed by

adding the blue and green angles.

m∠[blue] = 53

m∠[green] = 34

So x = [53] + [34].

53 + 34 = 87

So x = 87.