Marcus and Marlee are comparing △ABC with right angle B with △PQR with right angle Q to determine if they are congruent. AB¯¯¯¯¯¯¯¯ ≅ QR¯¯¯¯¯¯¯¯

Question

and AC¯¯¯¯¯¯¯¯ ≅ PR¯¯¯¯¯¯¯¯
. Marcus says the triangles are congruent by the HL Congruence Theorem, but Marlee does not agree. Who is correct? (Hint: Draw a picture.)(1 point)
Responses

Neither is correct; the triangles are congruent by Side-Side-Angle.
Neither is correct; the triangles are congruent by Side-Side-Angle.

Neither is correct; the triangles are congruent by Side-Angle-Side.
Neither is correct; the triangles are congruent by Side-Angle-Side.

Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle.
Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle.

Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent.

Answer 1

To determine whether Marcus or Marlee is correct, let's analyze the information given about triangles \( \triangle ABC \) and \( \triangle PQR \).

1. Key Information:

   • \( AB \cong QR \) (one leg of the right triangles is congruent)
   • \( AC \cong PR \) (the other leg of one triangle is congruent to the hypotenuse of the other triangle)

2. Congruence Theorems:

   • Hypotenuse-Leg congruence theorem (HL) states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
   • Side-Side-Angle (SSA) does not guarantee congruence and is thus not a valid test for this scenario.
   • Side-Angle-Side (SAS) requires two sides and the included angle to be congruent, which is not applicable here since one of the angles is not specified as being congruent.

3. Analysis:

   • The triangles are both right triangles.
   • Marcus claims they are congruent by the HL theorem, which requires that one leg and the hypotenuse of one right triangle be congruent to the corresponding parts of another right triangle.

Since we know the legs are congruent, we need to assess if the segments compared are indeed representing the hypotenuse and the leg correctly.

If we assume that \( AC \) and \( PR \) are indeed the hypotenuses of triangles \( ABC \) and \( PQR \) respectively, then Marcus is correct as the legs (one from each triangle) and the hypotenuses are congruent.

If Marlee assumes that no hypotenuse is indicated as being congruent, then she may assume there's insufficient information leading her to believe they can't be congruent.

However, following the information about sides given, if both a leg and the hypotenuse is congruent, then Marcus is indeed correct.

Conclusion: Marcus is correct; the triangles are right triangles where a leg and the hypotenuse of one triangle is congruent to a leg and hypotenuse of the other triangle.