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Marcus and Marlee are

comparing AABC with right angle B with APQR with right angle @ to determine if they are congruent.
AB = QR 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)
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.
Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent.
Neither is correct; the triangles are congruent by Mageiengle.

1 answer

To analyze the problem, let's first break down the congruence statements made by Marcus and Marlee.

  1. Given Information:

    • Triangles AABC and APQR are right triangles with right angles at B and P, respectively.
    • The sides given are:
      • \( AB = QR \)
      • \( AC = PR \)

  2. Marcus's Argument:

    • Marcus claims that the triangles are congruent by the HL (Hypotenuse-Leg) Congruence Theorem. The HL theorem states that if you have two right triangles, and the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of another triangle, then the triangles are congruent.

  3. Marlee's Argument:

    • Marlee does not agree with Marcus, suggesting the triangles are not necessarily congruent because the corresponding legs are not marked as congruent.

  4. Analysis: Given the information:

    • The hypotenuses of both triangles (AC and PR) can be inferred to be the longest sides of each triangle. Marcus is claiming congruency based on the given sides.
    • However, Marlee's reaction indicates a concern regarding whether the corresponding legs of the triangles have been shown or explicitly marked as congruent.

So, what's most important here is:

  • We know that for the HL theorem to apply, one leg and the hypotenuse of one triangle must be congruent to the corresponding leg and hypotenuse of the other triangle. Although legs \( AB \) and \( QR \) are given to be congruent and \( AC \) and \( PR \) are congruent, without explicitly stating the other leg (the one opposite \( B \) and \( P \)) as congruent, we cannot use HL to show the triangles are congruent.

Thus, since the information given does not specify the congruency of the second legs, Marlee's point stands. Therefore, we conclude:

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