Tests for parallel lines

A test for parallel lines lets us decide that two lines are parallel. Theorem: if two lines and a transversal form equal alternate interior angles, the two lines are parallel. Idea of the proof: let the transversal meet a and b at A and B. If the angles are right, a and b are both perpendicular to AB, hence parallel. Otherwise take the midpoint O of AB, drop the perpendicular OC to a and let it meet b at D; in △AOC and △BOD we have AO = BO, equal vertical angles and equal given angles, so △AOC = △BOD by ASA. Hence ∠C = ∠D = 90°, so a and b are both perpendicular to CD and therefore parallel. In practice, showing that alternate interior angles are equal proves the lines parallel.