Is trianglePQS congruent to triangleRQS by HL? If so, name the legs that allow the use of HL.
Triangle P S R is shown with altitude S Q from vertex S to point Q on side P R. Side P S is marked congruent to Side S R.
(2 points)
Yes, triangle PQS is congruent to triangle RQS by the Hypotenuse-Leg (HL) Postulate.
The legs that allow the use of HL are PS and QS.
To determine if triangle PQS is congruent to triangle RQS by the HL (Hypotenuse-Leg) congruence theorem, we need to check if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and corresponding leg of the other triangle.
Given that triangle PSR is shown with the altitude SQ from vertex S to point Q on side PR, and that side PS is congruent to side SR, we can deduce the following:
1. Hypotenuse HL: In triangle PQS and triangle RQS, the hypotenuse QS is common to both triangles.
2. Leg HL: In triangle PQS and triangle RQS, the leg PS of triangle PQS is congruent to the leg SR of triangle RQS.
Thus, triangle PQS is congruent to triangle RQS by the HL (Hypotenuse-Leg) congruence theorem, with the legs PS (of triangle PQS) and SR (of triangle RQS) being the legs that allow the use of HL.
To determine if triangle PQS is congruent to triangle RQS by the HL (Hypotenuse-Leg) criterion, we need to confirm if two conditions are satisfied:
1. The hypotenuse of one triangle is congruent to the hypotenuse of the other triangle.
2. One leg of one triangle is congruent to the corresponding leg of the other triangle.
Given that triangle PSR has altitude SQ from vertex S to point Q on side PR, and that side PS is congruent to side SR, we can determine whether the HL criterion applies.
First, let's identify the hypotenuses of the triangles. In this case, the hypotenuse is the side that is not part of the right angle. So, in triangle PQS, the hypotenuse is PS. Similarly, in triangle RQS, the hypotenuse is RS.
Now, we need to determine if PS is congruent to RS. You mentioned that side PS is congruent to side SR in the given diagram. Therefore, the hypotenuse of one triangle (PS) is congruent to the hypotenuse of the other triangle (RS). The first condition is satisfied.
Next, we need to identify the legs of the triangles. The leg for triangle PQS is PQ, and for triangle RQS, it is RQ.
To determine if the legs are congruent, we need additional information. Without any further provided information about the horizontal sides or any angles, we cannot conclude whether PQ is congruent to RQ. Hence, we do not have enough information to show that triangle PQS is congruent to triangle RQS through the HL criterion.