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**Drexel28** Let $\displaystyle \ell_1,\ell_2$ be the two lines. Translate $\displaystyle \ell_1$ over however many units so that it is not above $\displaystyle \ell_2$. Consider the projections $\displaystyle \pi:\ell_1,\ell_2\mapsto\mathbb{R}$. This is clearly a bijection. Thus, we must merely show that given two segments $\displaystyle (a,b),(c,d)$ that $\displaystyle (a,b)\simeq (c,d)$. Conjecture there exists a linear bijection between them. Then $\displaystyle f(x)=mx+k$. So then we need that $\displaystyle {f(a)=ma+k=d}\brace{f(b)=mb+k=d}$. 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?