1. ## Hyperbolic rigid motion

Let $AB$ and $CD$ be geodesic segments of equal length. Prove that there is a hyperbolic rigid motion that maps $A$ and $AB$ onto $C$ and $CD$, respectively.

Once again, completely lost.

2. Any suggestions?

3. Okay, so I know the cases will probably have to divided up depending on if the geodesic segments are vertical lines or arcs of circles centered on the x-axis, and there has to be some inversion(s), but after that point I am just not sure.

4. so you're using the upper plane model of the hyperbolic geometry, are you?

5. Yes, the upper half plane, so the hyperbolic rigid motions we have covered so far are inversions where the center is located on the x-axis, horizontal translations, and reflections across vertical lines.