From the diagram we know that \angle{G} and \angle{N} are both right angles making both triangles right triangles. We also know that \overline{MG} \cong \overline{NX} which is a pair of corresponding legs. We are missing congruent hypotenuses to complete the theorem.

\overline{MH}\cong \overline{XS}

Having a right angle alone is not enough to justify hypotenuse-leg. Make sure that the hypotenuses are congruent and one pair of legs are also congruent.

When proving triangles congruent, start by labeling and identifying the given information as well as anything that we can conclude directly from the diagram (such as vertical angles or reflexively congruent parts). Since we are trying to prove two triangles congruent, we need to decide which triangle congruence theorem best fits the given information. Then, when writing the proof we always start with the given information, build any conclusions based on the given information, and end with the conclusion we are trying to prove.

We are given that \overline{LK} \perp \overline{IJ} which means that \angle{IKL} and \angle{JKL} are both right angles by the definition of perpendicular lines. That means \triangle{ILK} and \triangle{JLK} are both right triangles by the definition of right triangles. We are also given that \overline{IL}\cong \overline{JL} which means we have two congruent hypotenuses, and we know that \overline{LK}\cong \overline{LK} by the reflexive property so we have a pair of congruent legs as well. This is enough information to conclude that \triangle{ILK}\cong \triangle{JLK} by the hypotenuse-leg theorem.

Triangle proofs can be written in any style - just make sure that each congruent part needed to justify the triangle congruence theorem has its own statement and reason included.