Cross ratio in hyperbolic geometry In the rough sketch four concurrent lines are drawn in the Poincaré disk model and in the Euclidean model.
If same angles $ (\alpha,\beta,\gamma,\delta) $  are enclosed at respective points of concurrency in either model then does the same trig definition of Cross Ratio hold good?

If that is so, and if the same three adjacent angles are given then is it correct to say they have the same Cross Ratio in euclidean and hyperbolic geometries?
I need your help, appreciate your comments.
 A: Yes.  The Poincaré disk model preserves angles everywhere, so we are free to (hyperbolically) translate the point of concurrency to the (Euclidean) center of the disk.  Then we can swap out the Poincaré disk model for the Beltrami–Klein model, which preserves projective invariants everywhere including the cross-ratio; the Beltrami–Klein model does not preserve angles everywhere, but it does preserve them at the center.
A: A convenient definition of the cross-ratio, in hyperbolic geometry, is as follows.
We work in the upper half plane model; see here, for example.  Suppose that $a$, $b$, $c$, and $d$ are points on the boundary of the upper half plane; that is, on the $x$-axis.  Then the cross-ratio of these four points (in order) is defined in the usual way; see here, for example.
The relevance for hyperbolic geometry is as follows.  The isometry group (a copy of $\textrm{PSL}(2, \mathbb{R})$) acts three-transitively on the the ideal boundary of hyperbolic space.  Thus for four-tuples of points of the boundary there is only one invariant - this is the cross ratio.
