"Prove: If two lines in the hyperbolic plane are asymptotic, then they do not admit a common perpendicular"
I'm trying to use something related to the angle of parallelism, but I guess it's not the right thing to do. Can someone suggest something? I feel kind of lost.
If you can use the relationship of parallelism and the concepts of pole and polar then the following figure will help:
Here $a$ and $b$ are ultra parallel lines so the line joining their poles is their common parallel. It is clear that if $a$ and $b$ gets asymptotic then the latter line "goes out" of the hyperbolic plane