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.

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:

Triangle \( \triangle ABC \) has a right angle at \( B \).
Triangle \( \triangle PQR \) has a right angle at \( Q \).
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.