Consider the following diagram. There are two parallel lines with two transversals which cross at arbitrary angles (ignore any symmetry in the drawing).
From the "alternate interior angles" theorem, we can conclude that 1 = 2, 3 = 4, 5 = 6, and 7 = 8 by considering each transversal separately. However, can we say something about pairs of angles formed by different transversals? For example, can we say that 6 = 1 and 2 = 5? Intuitively, it makes sense, but I'm having a hard time proving it.