Alternate Interior Angles: in Parallel Lines

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.

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.