In the Hyperbolic NonEuclidean World the size of a triangle is determined by interior angles. There is no similar triangle except congruence. Therefore dirction and position of the first triangle for reflection is optional, but you can not choose the size after three angles are set. Vertexes of ideal triangle are at infinity from the beginning. Any vertexes of other triangles never touch a point of infinity unless you lay out infinite number of them.
Picture below is four frames of a cartoon film with ideal triangles.
It is made by Makoto Sakuma. All triangles are congruent. Because all interior angles are zero and all sides are infinite. And they fully fill the model disc.