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.