Basic stategy Suppose that ABC and PQR are htriangles. Then there is a hyperbolic transformation t which maps A to P, B to B' on the hline PQ, on the same side of P as Q, and C to C' on the same side of the hline as R.
Note that, as t preserves angle and hyperbolic distance,



(AAA) condition If htriangles ABC and PQR have


Proof If any pair of sides have equal hyperbolic length, then the htriangles are hcongruent by the (ASA) condition. Assume that no two are equal (we will obtain a contradiction).
Let t be the transformation implied by the Basic Strategy.
By our assumptions, C' lies on PR, but C' ≠ R,
We have one of the two situations shown on the right.
