Given two triangles, if their hypotenuse are congruent, and one pair of their legs are congruent then the two triangles are congruent. In this post, we are going to prove this theorem.

In the figure below, ABC and DEF are right triangles with right angles at C and F, respectively.

It is given that and . We are going to prove that .

**Proof **

In , construct ray and mark so that (S). Since it is given (S) and (A), then by SAS Congruence, . Since corresponding parts of congruent triangles are congruent, .

It is given that , so by the transitive property of congruence. This means that is an isosceles triangle. Since the angles opposite of the congruent sides of a triangle are congruent, . Therefore, by AAS Congruence. Therefore, by the transitive property of congruence.