c What can you conclude about right triangles where a pair of legs and the hypotenuse are congruent?
D
d Do you have any information about the relationship between two corresponding sides of the triangles?
A
a x=6
Theorem: SAS Congruence Theorem
B
b x=10
Theorem: ASA Congruence Theorem
C
c x≈ 15.71
Theorem: HL Congruence Theorem
D
d Cannot be determined.
Practice makes perfect
a Examining the diagram, we see two triangles inside of △ ABC. Notice that two of these triangle's angles form a linear pair, which have been marked below.
The angles of a linear pair are supplementary, and because one angle is a right angle the second angle must be right as well.
m∠ θ+90^(∘)=180^(∘) ⇔ m∠ θ =90^(∘)
Also, notice that these right triangles share their vertical side. Therefore, this side is congruent as well according to the Reflexive Property of Congruence.
To claim congruence we also must show that at least one pair of corresponding sides is congruent. Notice that both triangles share BD as a side. Since this side is between the same two pairs of congruent angles, it shows two corresponding sides in our triangles.
c Examining the diagram, we see that △ ADC and △ CBD are both right triangles with one pair of congruent legs. They also share their hypotenuse, which means it is also congruent according to the Reflexive Property of Congruence.
d Examining the diagrams, we see that the triangles have two pairs of congruent angles.
With this information we can claim similarity by the AA Similarity Theorem. However, we do not have enough information about the side lengths. Therefore we do not have enough information to claim congruence, which means we cannot find x.
