Question
Marcus and Marlee are comparing △ABC with right angle at B with △PQR with right angle at Q to determine if they are congruent. Given AB≅QR and AC≅PR, Marcus says the triangles are congruent by the HL Congruence Theorem, but Marlee does not agree. Who is correct?
A. Neither is correct; the triangles are congruent by Side-Side-Angle.
B. Marlee is correct, the triangles are not congruent because the corresponding legs are not marked as congruent.
C. Neither is correct, the triangles are congruent by Side-Angle-Side.
D. 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.
A. Neither is correct; the triangles are congruent by Side-Side-Angle.
B. Marlee is correct, the triangles are not congruent because the corresponding legs are not marked as congruent.
C. Neither is correct, the triangles are congruent by Side-Angle-Side.
D. 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.
Answers
GPT-4o mini
In order to determine if triangles △ABC and △PQR are congruent using the given information, we should focus on the congruence theorems applicable to right triangles.
Recall that HL (Hypotenuse-Leg) Congruence Theorem states that if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
From the given information:
- AB ≅ QR (which is a leg of triangle ABC and a leg of triangle PQR)
- AC ≅ PR (which is the hypotenuse of triangle ABC and the hypotenuse of triangle PQR)
Marcus states that the triangles are congruent by the HL Congruence Theorem, which is valid because:
- We have one leg (AB ≅ QR) and the hypotenuse (AC ≅ PR) of both triangles congruent.
Marlee, on the other hand, does not think the triangles can be concluded as congruent, stating that the corresponding legs are not marked as congruent. However, she may not be recognizing that the congruences given (AB ≅ QR and AC ≅ PR) sufficiently meet the criteria for HL theorem.
Therefore, the correct answer is:
**D. 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.**
Recall that HL (Hypotenuse-Leg) Congruence Theorem states that if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
From the given information:
- AB ≅ QR (which is a leg of triangle ABC and a leg of triangle PQR)
- AC ≅ PR (which is the hypotenuse of triangle ABC and the hypotenuse of triangle PQR)
Marcus states that the triangles are congruent by the HL Congruence Theorem, which is valid because:
- We have one leg (AB ≅ QR) and the hypotenuse (AC ≅ PR) of both triangles congruent.
Marlee, on the other hand, does not think the triangles can be concluded as congruent, stating that the corresponding legs are not marked as congruent. However, she may not be recognizing that the congruences given (AB ≅ QR and AC ≅ PR) sufficiently meet the criteria for HL theorem.
Therefore, the correct answer is:
**D. 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.**