Let $A, B, C, D$ be four points with the following properties:
- Three of those points are never collinear.
- The angles $\angle ABC$ and $\angle DAB$ are right angles.
- $C$ and $D$ are located on the same side of the line running through $A$ and $B$.
- $AD \cong BC$.
I want to prove without use of the parallel postulate that the line running through $C$ and $D$ is parallel to the line running through $A$ and $B$. However, whenever I try to prove this, I really want to use the fact that the sum of the inner angles of every triangle equals two right angles (which of course assumes the parallel postulate). I think that it is pretty easy to show that $AD$ and $BC$ are parallel to each other, however that does not seem to be of a big help.
How should I approach this exercise? Should I prove it directly or by contradiction? Do I have to use the fact that $AD$ and $BC$ are parallel to each other? I really only need a hint.
Thanks for any answers in advance.
EDIT: I don't know whether this is relevant, but all of the other Hilbert axioms are given.