Definition: Alternate interior angles lie between two lines on opposite sides of a transversal; when the lines are parallel, these angles are congruent.
How do you spot them?
Look for two angles that are inside the pair of lines, on different sides of the transversal. Their shape forms a Z or a backward Z.
What is the theorem?
If two parallel lines are cut by a transversal, alternate interior angles are congruent. The converse is also true: if alternate interior angles are congruent, the two lines are parallel.
Example. Lines l and m are parallel. One alternate interior angle is (7x – 12) degrees and the other is (4x + 18) degrees. Setting them equal: 7x – 12 = 4x + 18, so 3x = 30 and x = 10. Each angle measures 58 degrees. If instead the measures were 58 and 60 degrees, the lines would not be parallel.
What do students get wrong?
They choose same-side interior angles, which are supplementary, not equal. Or they pick angles on the same side of the transversal. The word alternate means opposite sides. A good habit is to sketch the Z shape before writing the equation.