Question

1. which pair of triangles cannot be proven congruent using the hl theorem?

Explanation:

Step1: Recall HL - Hypotenuse - Leg theorem

The HL theorem states that if the hypotenuse and a leg of a right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent.

Step2: Analyze the first pair of triangles

In the first pair of triangles $\triangle{JKL}$ and $\triangle{MKL}$, $\angle{J}=\angle{M} = 90^{\circ}$, $KL$ is common (hypotenuse) and $JL=ML$ (a leg). So, they can be proven congruent by HL.

Step3: Analyze the second pair of triangles

In the second pair of triangles $\triangle{AST}$ and $\triangle{AUT}$, $\angle{T}=\angle{U}=90^{\circ}$, $AS = AU$ (hypotenuse) and $AT$ is common (a leg). So, they can be proven congruent by HL.

Since no non - congruent pair is shown in the problem statement and we assume the question is about the given pairs in general, if we had to pick a non - applicable case conceptually, we need to know that for HL to work, we must have right - triangles. If a pair of non - right triangles were given, they couldn't be proven congruent by HL. But since only right - triangle pairs are shown here, assuming there are other options not shown in the image, a pair of non - right triangles would be the answer. However, based on the given image, both pairs can be proven congruent by HL. If we assume the question has some error or missing parts and we go by the principle of HL requirements, a non - right triangle pair would be the non - applicable one for HL.

Answer:

There is no non - congruent pair shown in the given image for which HL cannot be applied. But conceptually, non - right triangles cannot be proven congruent using the HL theorem.