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

A
B
¯

≅

Q
R
¯
and AC¯¯¯¯¯¯¯¯ ≅ PR¯¯¯¯¯¯¯¯
A
C
¯

≅

P
R
¯
. 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

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.

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

Marlee is correct; the triangles are not congruent because the corresponding legs are not marked as congruent.
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 Side-Angle-Side.
Neither is correct; the triangles are congruent by Side-Angle-Side.

Answer 1

To determine whether Marcus or Marlee is correct, we need to analyze the given information and the conditions for triangle congruence.

1. Congruent Parts: Marcus states that \( AB \cong QR \) and \( AC \cong PR \). He then claims that the triangles are congruent by the HL (Hypotenuse-Leg) Congruence Theorem.

2. Right Triangles: Both triangles \( \triangle ABC \) and \( \triangle PQR \) are right triangles, with right angles at \( B \) and \( Q \) respectively.

3. HL Congruence Theorem: The HL theorem states that if in two right triangles, the hypotenuse and one leg of one triangle are congruent to the hypotenuse and one leg of the other triangle, then the triangles are congruent.

Now, we need to check if all conditions are satisfied for the HL theorem:

• The hypotenuse of triangle \( \triangle ABC \) (which is \( AC \)) must be congruent to the hypotenuse of triangle \( \triangle PQR \) (which is \( PR \)).
• Additionally, one of the legs of \( \triangle ABC \) must be congruent to one of the legs of \( \triangle PQR \).

Now, the critical point of uncertainty is whether the hypotenuse and leg pair being compared are indeed marked as congruent, or if the problem has indicated that the legs need to be considered. If only the legs are given as congruent and not the hypotenuses, then the triangles cannot be concluded to be congruent by the HL theorem.

Therefore, the most logical response would be:

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