[b]Theorem[/b]: If two triangles are similar in hyperbolic geometry, then they are congruent.[br][br][b]Proof Sketch[/b]: Suppose to the contrary that two hyperbolic triangles, [i][math]ABC[/math][/i] and [i][math]DEF[/math][/i], are similar but not congruent. [br][br]Then no sides are congruent, because otherwise ASA would hold (and it does in Hyperbolic Geometry). [br][br]Without loss of generality, assume [math]DE>AB[/math] and [math]DF>AC[/math].[br][br]Produce [i]E'[/i] on side [i]DE[/i] and [i]F'[/i] on side [i]DF[/i] so [math]DE'\cong AB[/math] and [math]DF'\cong AC[/math].[br][br]Then quadrilateral [i]EE'F'F[/i] has angle sum [math]2\pi[/math] which violates the angle sum theorem.□

Why does this proof not work in Euclidean Geometry?

[b]Corollary[/b]: If two regular polygons are similar in Hyperbolic Geometry, then they are congruent.[br][br][b]Proof[/b]: Triangulate and apply the previous theorem. □