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¯¯¯¯¯¯¯¯≅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¯¯¯¯¯¯¯¯ Modifying above upper B upper C with bar congruent to Modifying above upper X upper Z with bar AB¯¯¯¯¯¯¯¯≅YZ¯¯¯¯¯¯¯

1 answer

To show that triangles ΔABC and ΔXYZ are congruent by the HL (Hypotenuse-Leg) Congruence Theorem, we need to establish that the two triangles share two specific pieces of information:

  1. The hypotenuses \( AC \) and \( XZ \) are congruent (which you stated is true).
  2. The legs of the triangles corresponding to the acute angles adjacent to the hypotenuse must also be congruent.

Since angle \( B \) and angle \( Y \) are congruent, we are now looking for the congruence of one leg from each triangle.

Given that we need to find a corresponding leg congruence, the appropriate corresponding parts to show would be:

AB ≅ XY or BC ≅ YZ.

Both of these options present valid correspondences, but we should follow the typical notation of the HL Theorem. The correct choice here would depend on the arrangement of points in the triangles.

For the context given in the options, one could select either of the following as suitable to justify HL:

  • AB ≅ XY or
  • BC ≅ YZ

Typically, we'd go with the first option, which would set us up correctly under the HL theorem.

So one complete statement can be:

  • AB ≅ XY

This means that with the pairs of congruent sides and the congruence of the angles, we can conclude \( ΔABC ≅ ΔXYZ \).