How to use the hl congruence theorem to determine if the given triangles are congruent

The HL theorem works only for right-triangles.

and if the right-triangles have equal hypotenuses and one of the other sides, then the triangles are congruent by HL.

now, from the picture, we can see that PQ = ST, ok, what about the hypotenuses?  are they equal?  let's check by setting them equal to each other, and let's see what value of "x" makes them so.



the system of equations for both hypotenuses, do not yield a feasible "x" value, and thus is inconsistent or has no solution, meaning there's no value of "x" that wil make that true.

therefore, the hypotenuses can never be equal, and thus the triangles aren't congruent.