I know that if two lines are parallel and there is a transversal crossing both, the alternate interior angles are congruent, alternate exterior angles congruent, etc. etc.
According to the geometry textbook that the student I'm tutoring brought, the converse is true as well: if the alternate interior angles are congruent, or the consecutive interior angles are supplementary, etc., then the lines must be parallel.
Why should this be? I mean, it looks like it should be true, but what is the proof that it's true?
Assuming you are working from some variation of Euclid's axioms, you can prove this as follows. Suppose two lines $\ell$ and $m$ are not parallel but form congruent alternate interior angles with a third line $n$. Since $\ell$ and $m$ are not parallel, we can thus form a triangle whose sides are segments of $\ell$, $m$, and $n$. But by hypothesis, the two angles of this triangle along the segment of $n$ are supplementary. This is a contradiction, since two angles in a triangle must add up to less than $\pi$.
(Note that some care is needed to avoid circularity in this last step: the proofs that I know of for the fact that the sum of the angles in a triangle is $\pi$ use the existence of parallel lines, and the classical construction of parallel lines uses the statement we are trying to prove here. However, the statement that two angles in a triangle add up to less than $\pi$ can be proved without using parallel lines; see for instance Euclid Book I, Proposition 17 which is a corollary of Proposition 16. Alternatively, in axiomatizations of geometry that include a Dedekind-style completeness axiom, the existence of parallel lines can be proved from completeness.)