Question

Can someone please check this real quick?

Is triangle PQS congruent to triangle RQS by HL? If so, name the legs that allow the use of HL.

My answer:
"The Hypotenuse-Leg (HL) Theorem is when a leg and the hypotenuse in one triangle are congruent to the corresponding parts in another triangle. Two right triangles make up the triangle that serves as the diagram for this question. PR is congruent to RQ. Those are the legs. PS and RS, or the side with the tick mark intersecting is the hypotenuse."

Am I correct?

Answer 1

Drawing is not to scale.

Answer 2

From the picture I am assuming you've been given that PS = RS. If so that is your hypotenuse. Now to choose a leg the clearest and easiest one to prove is that SQ = SQ by reflection. I am also assuming that you've been told SQ is an altitude so you can say angle sqp = angle rqp = 90 by definition of altitude. That is enough to finish your proof.

Answer 3

What HL theorem really is permission to use SSA congruency which is only allowed when you are dealing with right triangles.

Answer 4

No, I have not been given the information of PS = RS, and SQ is an altitude.

Answer 5

@hoodymay

Answer 6

Did the original drawing show the tick marks and the square symbol at the bottom of the altitude? These mean the sides with the ticks are congruent to eachother, and that the line SQ is an altitude? If so the previous answer is ok.

Answer 7

What do you mean by the altitude?

Answer 8

An altitude in a line from a side of the triangle to an opposing vertex which forms a 90 degree angle at the side. In your picture above line SQ goes from side PR to vertex S and the little symbol in your picture implies that SQ is perpendicular to side PR. So the two smaller triangles are right triangles in this picture.

Answer 9

Okay, I understand. So...
"Line segments PR and RS are congruent to one another. This is the hypotenuse. SQ is congruent to SQ by the Reflexive Property. Since two right triangles make up the triangle in the diagram...angle SQP is congruent to angle RQP which are 90 degrees by the definition of altitude." Is this correct and sufficient?

Answer 10

@hoodymay

Answer 11

Yes exactly. So your two triangles are congruent by HL theorem in a right triangle.

Answer 12

Okay, thank you very much!

Answer 13

No problem :)

Answer 14

No the PQS = RQS is not by HL.

Answer 15

HL stands for Hypotenuse Leg. It states "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 right triangles are congruent."

Since PS and RS are congruent, and both triangles share the same hypotenuse, then both triangles are congruent. They're congruent by the sides PS, RS, and the hypotenuse QS.