Proof: Hypotenuse-Leg Criterion
Proof : Hypotenuse-Leg Criterion
Given two triangles, ABC and DEF, with right angles A and D, AB DE, and BC EF. Prove that ABC DEF. This is the Hypotenuse-Leg criterion for congruence of right triangles.
Note: If AC DF, then these triangles would be congruent because of SAS:
(we would have AB DE, A D, and AC DF)
Proof: Assume AC is not congruent to DF. By (III-1), there is a point X on ray DF such that AC DX.
By our assumption, we know that X is different from F. We know that ABC DEX because
AB DE, A D, and AC DX. This implies that BC EX but this is not possible because
BC EF by our assumption. Thus, X must be point F. Therefore, ABC DEF.