Question

Given: Angle R is congruent to. Angle T, PQ is parellel to ST, and PQS is congruent to R Prove: Triangle PQS is congruent to Triangle STP(ignore the markings on the figures)

Answer 1

If PQ is parallel to ST, the angle ∠PST is alternate interior to the angle ∠QPS, so they are congruent.

Also, we have that ∠R = ∠T = ∠PQS.

If the angles ∠PQS and ∠T are congruent and PQ is parallel to ST, the figure PQST is a parallelogram (because QS and PT will also be parallel), so the sides PQ and ST have the same length, and QS and PT have the same length as well.

Then, we have that the triangles PQS and STP are congruent by the case SAS (sides PQ-ST, angles ∠PQS and ∠T, sides QS and PT).