consider right triangles triangle ABC and triangle XYZ. Angle B and angle Y are congruent. the hypotenuse of each triangle is line AC and line XZ, and they are congruent. what other corresponding parts must be congruent to show that triangle ABC = triangle XYZ by the HL congruence theorem?

Question

consider right triangles triangle ABC and triangle XYZ. Angle B and angle Y are congruent. the hypotenuse of each triangle is line AC and line XZ, and they are congruent. what other corresponding parts must be congruent to show that triangle ABC = triangle XYZ by the HL congruence theorem?
A. BC=XZ
B. AB=XY
C.AB=YZ
D. BC=XY

GPT-4o mini · Answer

To use the Hypotenuse-Leg (HL) congruence theorem for proving that triangle ABC is congruent to triangle XYZ, we need to confirm that the hypotenuses are congruent and one pair of legs is also congruent.

From the information you provided:

The hypotenuses AC and XZ are congruent.
Angles B and Y are congruent.

Now, we need to choose one of the options that represent the legs of the triangles. The HL theorem requires that one corresponding leg of each triangle be equal along with the hypotenuse.

The legs in triangle ABC are AB and BC, and the legs in triangle XYZ are XY and YZ. Therefore, we need one pair of legs to show congruence:

Option A: BC = XZ (This does not work because BC is a leg of triangle ABC, and XZ is the hypotenuse of triangle XYZ.)
Option B: AB = XY (This is a leg of triangle ABC, and corresponds to a leg in triangle XYZ.)
Option C: AB = YZ (This is a leg of triangle ABC, but YZ is not a leg of triangle XYZ.)
Option D: BC = XY (This is a leg of triangle ABC, but again, XY is the hypotenuse of triangle XYZ.)

Since we need to show that one leg of triangle ABC is congruent to one leg of triangle XYZ, the correct option would be:

B. AB = XY

This shows that one leg in each triangle is congruent along with the congruence of the hypotenuses, satisfying the conditions of the HL theorem.