Given: In the figure, RT= RS, PQ // RS. Prove PQ= PT. Kindly use Two-column proof.

Question

Answer 1

Hey Beaux, here is what I came up with noting that both the line through R and T and the line through S and T are transversals of the parallel lines through P and Q, and R and S:

1. ∠TPQ ≅ ∠ TRS (These are corresponding angles in a transversal system).
2. ∠TQP ≅ ∠TSR (These are corresponding angles in a transversal system).
3. ΔTRS ∼ ΔTPQ (By lines 1 and 2 and the angle-angle similarity theorem).
4. PQ/RS = PT/RT (Corresponding sides of similar triangles).
5. PQ/RS = PT/RS (RS=RT).
6. PQ = PT (By line 5).

Hope this helps!