The (SAS) condition for hyperbolic triangles

Basic stategy
Suppose that ABC and PQR are h-triangles.
Then there is a hyperbolic transformation t which maps
A to P,
B to B' on the h-line PQ, on the same side of P as Q, and
C to C' on the same side of the h-line as R.

Note that, as t preserves angle and hyperbolic distance,
d(A,B) = d(P,B'), d(A,C) = d(P,C'), d(B,C) = d(B',C'), and
<ABC = <PB'C', <BAC = <QP'C', <ACB = <PC'Q.

(SAS) condition
If h-triangles ABC and PQR have
(1) d(A,B) = d(P,Q),
(2) <BAC = <QPR, and
(3) d(A,C) = d(P,R),
then the h-triangles are h-congruent.

Proof
Let t be the transformation implied by the Basic Strategy.
Then B' lies on the h-line PQ.
By the Basic Strategy, d(A,B) = d(P,B').
By (1), d(A,B) = d(P,Q), so d(P,B') = d(P,Q), and hence B' = Q.
By the Basic Strategy, <BAC = <QPC'.
By (2), <BAC = <QPR, so C' lies on QR.
By the Basic Strategy, d(A,C) = d(P,C').
By (3), d(A,C) = d(P,R), so d(P,C') = d(P,R), and hence C' = R.
Thus t maps ABC to PQR, so the h-triangles are h-congruent.