Proof of Hypotenuse-Leg Criterion for congruence of right triangles

"Given two triangles,

and

with right angles at

and

, and

. Prove that

." Proof: Consider the point X, on the line

such that X is on the opposite side of A from C and such that

. By Euclid's Proposition 13,

is a right angle. Thus,

. We also know that

. Therefore, we have satisfied the SAS criterion for congruent triangles. That is,

. Notice then, that

. Thus,

is an isosceles triangle. Therefore,

Then, the AAS criterion for congruent triangles is satisfied for

and

. So, we can conclude that

New Resources