If we can show that △ QPR ≅ △ TPS, then we know that ∠Q ≅ ∠T. Examining the diagram, we see that ∠QPR and ∠TPS forms vertical angles. According to the Vertical Angles Congruence Theorem, vertical angles are congruent. Let's mark this in our diagram.
Since we know two pairs of congruent corresponding sides and included angle, we can by the SAS Congruence Theorem prove that △ QPR ≅ △ TPS. Now we know that ∠Q ≅ ∠T since these are corresponding angles.
