To work with the two larger triangles â–³ PUX and â–³ QSY, we might have to first consider the two pairs of smaller triangles that can be identified in the diagram.
See solution.
Practice makes perfect
To prove that △ PUX ≅ △ QSY, we start by separating the two triangles.
Having separated the triangles, we notice that △ PRU and △ QVS have two pairs of congruent corresponding angles, ∠PRU≅ ∠QVS and ∠PUR≅ ∠QSV If we can show that the triangles also have a congruent corresponding side, we have enough information to prove congruence. Let's look at the triangles when they are not separated.
Using the Segment Addition Postulate, we can write two equations:
&RU=RS+SU
&VS=VU+SU
Since SU is shared between the triangles, the Reflexive Property of Congruence tells us that this side is congruent. Also, we know that RS≅ VU which means we can use the Substitution Property of Equality to rewrite the first equation:
&RU= VU+SU
&VS=VU+SU
Finally, by the Transitive Property of Equality, we are able to show that RU≅ VU:
RU=VS ⇔ RU≅ VS
Now we have enough information to prove that △ PRU ≅ △ QVS by the ASA Congruence Theorem. Let's separate the triangles once more.
From the diagram, we see that ∠URP and ∠URX form a linear pair and so do ∠SVQ and ∠SVY. By the Linear Pair Postulate we know these two pairs of angles are supplementary. Since ∠SVQ≅ ∠URP, we can by the Congruence Supplements Theorem prove the following.
∠URX≅ ∠SVY
Let's add this information to the diagram.
Now we have enough information to prove that △ URX ≅ △ SVY by the ASA Congruence Theorem. Finally, we will mark all congruent parts in the two triangles.
From the diagram above, we have enough information to prove congruence between â–³ PUX and â–³ QSY. Let's remove all unnecessary parts and only focus on the information we need.
