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 AB¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper A upper B with bar congruent to Modifying above upper X upper Y with bar BC¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper B upper C with bar congruent to Modifying above upper X upper Y with bar BC¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯

1 answer

To demonstrate that triangles \( \Delta ABC \) and \( \Delta XYZ \) are congruent by the Hypotenuse-Leg (HL) Congruence Theorem, you already know that:

  1. The hypotenuses \( AC \) and \( XZ \) are congruent, which is given.
  2. Angles \( B \) and \( Y \) are congruent, which is also given.

The HL Theorem specifically requires that the hypotenuse and one leg of each triangle are congruent. Since the hypotenuse is already matched, we need to establish congruence of one of the legs.

From the options provided, we can analyze them:

  • \( AB \cong YZ \): Here, this option establishes that the leg \( AB \) corresponds to \( YZ \).
  • \( AB \cong XY \): This establishes that the leg \( AB \) corresponds to \( XY \).
  • \( BC \cong XY \): This establishes that the leg \( BC \) corresponds to \( XY \).
  • \( BC \cong XZ \): This option does not work because \( XZ \) is the hypotenuse, not a leg.

Since angle \( B \) and angle \( Y \) are already known to be congruent, establishing congruence of either \( AB \) or \( BC \) with one of the legs in \( \Delta XYZ \) (either \( YZ \) or \( XY \)) would suffice for congruence.

Choosing one of the leg options, we can conclude that:

  • \( AB \cong YZ \) (or \( AB \cong XY \) or \( BC \cong XY \))

Thus, you can select either \( AB \cong YZ \) or \( AB \cong XY \) to satisfy the HL theorem, concluding that:

\[ \Delta ABC \cong \Delta XYZ \]

For your specific case, you could choose:

AB¯¯¯¯¯¯¯¯≅YZ¯¯¯¯¯¯¯

as the necessary correspondence to show that \( \Delta ABC \cong \Delta XYZ \) by the HL Congruence Theorem.