Consider right triangles ΔABC and ΔXYZ. Angle B and angle Y are congruent. The hypotenuse of each triangle is AC¯¯¯¯¯¯¯¯ and XZ¯¯¯¯¯¯¯¯, and they are congruent. What other corresponding parts must be congruent to show that ΔABC≅ΔXYZ by the HL Congruence Theorem?(1 point) Responses AB¯¯¯¯¯¯¯¯≅YZ¯¯¯¯¯¯¯ Modifying above upper A upper B with bar congruent to Modifying above upper Y upper Z with bar BC¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯ Modifying above upper B upper C with bar congruent to Modifying above upper X upper Z with bar AB¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper A upper B with bar congruent to Modifying above upper X upper Y with bar BC¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯

Question

Select One

Answer 1

To show that triangles ΔABC and ΔXYZ are congruent by the Hypotenuse-Leg (HL) Congruence Theorem, we need to have the following:

1. The hypotenuses AC and XZ are congruent (which is given).
2. One leg of each triangle must also be congruent.

Since angle B and angle Y are congruent and the hypotenuses are given as congruent, we look for a pair of legs that must be congruent to apply HL.

By inspecting the options you've provided:

1. AB ≅ YZ - This doesn't relate the congruent legs correctly as YZ is not the other leg corresponding to B.
2. BC ≅ XZ - This is incorrect because XZ is the hypotenuse, not a leg.
3. AB ≅ XY - This suggests that one leg of triangle ABC (AB) is congruent to one leg of triangle XYZ (XY), which is a necessary condition.
4. BC ≅ XY - This indicates BC is congruent to XY, but does not include the required leg assumption needed.

The correct response to establish ΔABC ≅ ΔXYZ by the HL Congruence Theorem would be:

AB ≅ XY

This choice shows that one leg of triangle ABC is congruent to one leg of triangle XYZ, in addition to the congruence of the hypotenuses, validating the application of the HL theorem.