I've studied the hyperbolic plane by analytically building up the hyperboloid model, the Klein—Beltrami disc, the Poincaré disc, and the half-plane model from scratch. Now I'd like to prove that, given $\alpha,\beta,\gamma>0$ with $\alpha+\beta+\gamma<\pi$, there exists a hyperbolic triangle with angles $\alpha$, $\beta$ and $\gamma$.
Is there an easy proof of this fact? I mean a proof that only starts from the basic Riemannian structure of the models: the knowledge what the geodesics are, an expression for the distance, the fact that some models are conformal. (Of course, I have found proofs in literature, but they tend to refer a lot to other results, requiring e.g. an extensive theory of trigonometry.)