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

1 answer

To show that triangles ΔABC and ΔXYZ are congruent (ΔABC ≅ ΔXYZ) using the HL (Hypotenuse-Leg) Congruence Theorem, we need to establish that:

  1. The hypotenuses are congruent. Given that \(AC \cong XZ\), this condition is satisfied.
  2. One leg of the triangles must also be congruent. We need to show that either side \(AB\) or \(BC\) is congruent to the corresponding side in triangle \(XYZ\).

Since angle B and angle Y are congruent and the hypotenuses (AC and XZ) are congruent, we can state that the other leg must be either \(AB\) or \(BC\) congruent to a leg in triangle \(XYZ\).

The two possible responses for the legs that could be congruent are:

  • AB ≅ YZ (If we take the leg opposite angle B in triangle ΔABC)
  • BC ≅ XY (If we take the leg adjacent to angle B in triangle ΔABC)

However, the conditions state one angle and one hypotenuse are congruent. Therefore, we would need to show:

  • AB ≅ YZ or BC ≅ XY for a complete proof using HL.

Among your options, you would take the first leg option:

AB ≅ YZ.

This would satisfy the requirement that at least one corresponding leg is congruent to prove ΔABC ≅ ΔXYZ by the HL Congruence Theorem.