I have done quite a bit of hyperbolic geometry and euclidean geometry, but one thing remained obscure to me throughout: the connection between axioms and the metrical definition of geometry.
As I learned it, hyperbolic geometry was constructed as a counterexample to the fact that the Parallel Postulate can be derived from the previous Euclidean axioms. This makes sense - clearly, given a line and a point, infinitely many non-intersecting lines through that point can be constructed.
But then I thought about the all right angles are equal axiom. Does this mean that all right triangles are isomorphic? If yes, doesn't this require a notion of distance on the Euclidean plane?
Given the hyperbolic infinite and spherical zero number of parallel lines, I soon wondered if there were geometries with exactly $2$ or $k$ lines parallel to a given line and passing through a point? To solve this problem what constitutes, formally, a geometry/ how far can the term be reduced? I have seen definitions along the lines of $(P,l)$ for a point set P and line set $l$, but I do not see how this definition could be applied to solving the problem above.