Consider the HL Theorem and construct your proof based on this theorem.
a: The definition of supplementary angles b: The definition of right triangle c: Given d: Reflexive property of congruence e: The HL Theorem
Practice makes perfect
We have been given a flow proof and need to fill in the blank spaces. In order to do this, we will examine the given figure and information. Then we will make sure that the statements in the blanks allow for the desired outcome of the proof.
Given:& PS≅ PT, ∠ PRS ≅ ∠ PRT
Prove:& △ PRS ≅ △ PRT
Let's take a look at the statements in the proof.
Missing Information a
Let's begin by finding the missing information for blank a. We know that ∠ PRS ≅ ∠ PRT is given and they are supplementary angles. Therefore, we can complete the blank using the definition of supplementary angles.
∠ PRS and ∠ PRT are right angles.
a. Definition of supplementary angles
Let's continue!
Missing Information c and d
As we can see, the statement for the missing information c is the part of given information. We can also complete the blank d by using the Reflexive Property of Congruence.
PS≅ PT & PR≅ PR
c. Given & d. Reflexive Property of
& Congruence
Missing Information b
Since △ PRS and △ PRT have right angles, we can say that they are right triangles by using the definition of right triangle.
△ PRS and △ PRT are right triangles.
b. Definition of a right triangle
Missing Information e
Finally, we have everything we need to finish the proof! As we can see, the hypotenuse and one leg of △ PRS are congruent to the hypotenuse and one leg of △ PRT. In this case, we can use the Hypotenuse-Leg (HL) Theorem.
Hypotenuse-Leg (HL) Theorem
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
The illustration of the theorem is given below.
If △ PQR and △ XYZ right triangles,
PR≅ XZ, and PQ≅ XY,
then △ PQR≅ △ XYZ.
According to the theorem, we can finish the proof as follows.
△ PRS and △ PRT
e. HL Theorem
