Question

Marcus and Marlee are comparing △ABC with right angle B with △PQR with right angle Q 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) Responses Neither is correct; the triangles are congruent by Side-Side-Angle. Neither is correct; the triangles are congruent by Side-Side-Angle. 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-Angle-Side. Neither is correct; the triangles are congruent by Side-Angle-Side. 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 regarding the congruence of triangles \( \triangle ABC \) and \( \triangle PQR \), let's analyze the information given:

1. **Triangle \( \triangle ABC \)** has a right angle at \( B \).
2. **Triangle \( \triangle PQR \)** has a right angle at \( Q \).
3. It is stated that \( \overline{AB} \cong \overline{QR} \) (one leg of each triangle is congruent) and \( \overline{AC} \cong \overline{PR} \) (the hypotenuse of each triangle is congruent).

**Marcus's Argument:** He claims the triangles are congruent by the Hypotenuse-Leg (HL) Congruence Theorem since he has one leg and the hypotenuse of each triangle congruent.

**Marlee's Argument:** She disagrees, presumably because the corresponding legs are not indicated as congruent, which could imply that they do not have enough information to use HL.

**Analysis:**

- The Hypotenuse-Leg theorem states that if you have a right triangle, having congruent hypotenuses and one congruent leg means the triangles are congruent.
- Since it is given that \( \overline{AB} \cong \overline{QR} \) and \( \overline{AC} \cong \overline{PR} \), and since both triangles are right triangles (with respective right angles at \( B \) and \( Q \)), they indeed satisfy the conditions of the HL theorem.

Therefore, **Marcus is correct** in stating that the triangles are congruent by the HL Congruence Theorem.

In conclusion, the correct response is:

**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.**