Let

be the two lines. Translate

over however many units so that it is not above

. Consider the projections

. This is clearly a bijection. Thus, we must merely show that given two segments

that

. Conjecture there exists a linear bijection between them. Then

. So then we need that

. Solving this system of equations will give you a bijective function between the two which is actually a homeomorphism (that last part isn't necessary)

And, I don't see how defining an equivalence relation helps.

Wait....I read the question. But now I am having second thoughts about it's meaning. I thought you meant to show that any two lines has the same number of lines ("equivalent" in the category of sets). Is there somethign else?