EDIT1:

In the rough sketch three concurrent lines are cut ( or may not be shown cut) by a fourth transversal $T$ in the Poincaré disk model representation (single intersection) :

[![enter image description here][1]][1]


If angles $ \alpha,\beta,\gamma,\delta$  are drawn at point of concurrency of lines $(1,2,3)$ in the disk model. Does the trig definition of Cross Ratio hold good here as well?

  [1]: https://i.sstatic.net/5dOFN.png