Triangle PQT is congruent to triangle RST based on the AAS Congruence theorem. See the two column proof below.
What is a two column proof?
A two-column proof is a method of organizing and presenting a mathematical argument in a clear and structured format, with statements written in one column and corresponding reasons or justifications in an adjacent column..
The two column proof for the information given can be written as follows:
Statement Reason
1. PQ⊥QS, RS⊥SQ, 1. Given
T is the midpoint of PR
2. ∠Q ≅ ∠S 2. Def. of perpendicular lines
3. PT ≅ RT 3. Def. of midpoint
4. ∠QTP ≅ ∠STR 4. vertical angles theorem
5. ΔPQT ≅ ΔRST 5. AAS Congruence theorem