It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles.

Question. Is an analogous equidecomposability result true in the hyperbolic plane? And if yes, how such equidecomposability can be fulfilled?

In the Euclidean plane the equidecomposability is proved by reducing the problem to the equidecomposability of a triangle and a rectangle with one side of length 1. But the hyperbolic plane contains no rectangles, so this method does not work. Then, which method does work in the hyperbolic plane?