# The Right Triangle Leg Leg Congruence Theorem

The Right Triangle Leg Leg Congruence Theorem is a theorem for right triangles that says if the two corresponding shorter legs of two right triangles are congruent, then the two triangles are congruent. For instance, if the length of the shorter legs of the two triangles is 5, and the other leg is 12, then we are sure that the two triangles are congruent. Yes, they are in fact the triangles with side lengths 5, 12 and 13.

Theorem

Given two triangles $MNO$ and $PQR$ right angled at $N$ and $Q$ respectively.  If $\overline{MN} \cong \overline{NQ}$and $\overline{NO} \cong \overline{QR}$, then triangle $MNO$ is congruent to triangle $PQR$.

Proof

$\overline{MN} \cong \overline{NQ}$ (Given)

$\angle MNO \cong \angle PQR$ (All right angles are congruent, Postulate 4).

$\overline{NO} \cong \overline{QR}$ (Given)

Therefore, by SAS Triangle congruence, triangle $MNO$ and triangle $PQR$ are congruent.

Note that the Leg Leg Congruence Theorem is also related to the Hypotenuse-Leg Theorem.