Marcus and Marlee are comparing two right triangles to determine if they are congruent. Marcus has △ABC with a right angle at B . Side AB¯¯¯¯¯¯¯¯ has 2 tick marks, and the hypotenuse has 1 tick mark. Marlee has △PQR with a right angle at Q . Side QR¯¯¯¯¯¯¯¯ has 2 tick marks, and the hypotenuse has 1 tick mark. Marcus says the triangles are congruent by the HL Congruence Theorem, but Marlee does not agree. Who is correct
Marcus is correct. The HL (Hypotenuse-Leg) Congruence Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent. In this case, both triangles have a leg with 2 tick marks and a hypotenuse with 1 tick mark, so they satisfy the criteria for the HL Congruence Theorem. Therefore, the triangles are congruent.
To determine if the triangles are congruent, we need to check if they satisfy the conditions of the HL Congruence Theorem.
The HL Congruence Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
In this case, Marcus claims that △ABC is congruent to △PQR using the HL Congruence Theorem. He states that the hypotenuse and one leg of △ABC are congruent to the hypotenuse and one leg of △PQR.
Let's check if Marcus is correct.
The hypotenuse of △ABC has 1 tick mark, and the hypotenuse of △PQR also has 1 tick mark. So, the hypotenuses of both triangles are congruent.
One leg of △ABC has 2 tick marks, but we don't know the length of the leg of △PQR.
Therefore, Marcus's claim is incomplete and cannot be supported with the given information.
Marlee, on the other hand, is correct in saying that we do not have enough information to determine if the two triangles are congruent. We cannot use the HL Congruence Theorem without knowing the length of the leg of △PQR.
To determine who is correct, we need to apply the HL Congruence Theorem to the given information on the triangles.
The HL Congruence Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
Here's how Marcus and Marlee can use the HL Congruence Theorem to compare the triangles:
Marcus claims that △ABC and △PQR are congruent using the HL Congruence Theorem. According to Marcus, side AB and side QR are congruent because they both have 2 tick marks, and the hypotenuses of both triangles have 1 tick mark, making them congruent as well.
Marlee, on the other hand, disagrees with Marcus. To verify her point of view, let's compare the triangles based on the HL Congruence Theorem. Marlee says that to apply the HL Congruence Theorem, both the hypotenuse and leg of one triangle must be congruent to the corresponding parts of the other triangle.
Upon examining the given information, we find that Marcus and Marlee have only identified the congruency of the hypotenuses and one leg. They haven't compared the other leg of both triangles.
To determine the congruence of the entire triangles, we need to check the congruence of the remaining legs. However, the length of side BC and side PQ is not given, neither through tick marks nor any other measurements.
Without information about the congruence of the remaining leg, we cannot accurately determine if △ABC and △PQR are congruent.
In this case, Marlee is correct in stating that there isn't enough information to conclude congruence between the triangles based on the HL Congruence Theorem.