alternate interior angles
When a transversal crosses two lines, four of the eight angles lie in the interior — the region between the two lines — and four lie in the exterior. The alternate interior angles are the interior pair on opposite sides of the transversal. They are often spotted by the shape of a letter Z: the two angles cradled inside the bends of the Z are alternate interior angles. (The mirror pairs outside the two lines are the alternate exterior angles.)
Precisely: of the four interior angles, take one at the upper crossing and one at the lower crossing that sit on opposite sides of the transversal — these form an alternate-interior pair, and there are two such pairs. The alternate-interior-angles theorem says that if the two crossed lines are parallel, the alternate interior angles are equal. The quick proof: a corresponding angle at the far crossing equals the near one (corresponding angles), and that corresponding angle is also vertical to the alternate interior angle, hence equal — chaining the two gives the result. The converse also holds: equal alternate interior angles force the lines to be parallel.
Alternate interior angles are the workhorse of parallel-line proofs, and they appear constantly in the proof that a triangle's angles sum to 180: drawing a line through one vertex parallel to the opposite side turns the triangle's two base angles into alternate interior angles with the parallel, sliding them up to the apex.
A transversal cuts two parallel lines. One interior angle on the left of the transversal at the top crossing is 64 degrees. Its alternate interior angle, on the right of the transversal at the bottom crossing, is also 64 degrees.
With parallel lines, alternate interior angles are equal (the Z-shape).
Like corresponding angles, alternate interior angles are equal only when the lines are parallel; the matching pairs outside the two lines are alternate exterior angles, which behave the same way.