If we examine the point where PR and SQ intersect, we can identify two pairs of vertical angles. According to the Vertical Angles Congruence Theorem, such angles are congruent. Let's mark these in our diagram.
Now we have enough information to prove that
△ PTS ≅ △ RTQ and △ PTQ ≅ △ RTS,
by the SAS Congruence Theorem.
â–ł PRS and â–ł RPQ
If we view PR as a transversal to the parallel sides SP and RQ, as well as to the second pair of parallel sides, PQ and SR, we can identify two pairs of congruent corresponding angles by the Alternate Interior Angles Theorem.
Note that both triangles share PR as a side. By the Reflexive Property of Congruence, we can claim that this side is congruent in our triangles. By adding all of this information to our diagram, we can more easily see why these triangles are congruent.
Now we have enough information to prove that
△ PRS ≅ △ RPQ
by the ASA Congruence Theorem.
â–ł PQS and â–ł RSQ
If we view SQ as a transversal to the parallel sides SP and RQ, as well as to the second pair of parallel sides, PQ and SR, we can identify two pairs of congruent corresponding angles by the Alternate Interior Angles Theorem.
Like before, we see that both triangles share SQ as a side. By the Reflexive Property of Congruence, we can claim that this side is congruent in our triangles. By adding all of this information to our diagram, we can more easily see why these triangles are congruent.
Now we have enough information to prove that
△ PQS ≅ △ RSQ
by the ASA Congruence Theorem.
