To prove that triangles $\triangle RQT$ and $\triangle PQS$ are congruent using the SAS (Side-Angle-Side) criterion, we need two pairs of congruent sides and a pair of congruent angles between those sides.

1. Sides: We know that $RQ \cong QP$ and $RT \parallel SP$ are given. Since $Q$ bisects $ST$, we also know $TQ \cong QS$.

2. Angles: For the angles, we need $\angle RQT \cong \angle PQS$. The correct reason to use here is the Vertical Angles Theorem, as the angles $\angle RQT$ and $\angle PQS$ are formed by intersecting lines at point $Q$, making them vertical angles.

Thus, the correct answer is vertical angles theorem.

Would you like further details on any part of this proof?

Here are 5 follow-up questions:

1. Why is the vertical angles theorem essential in this proof?
2. How does the SAS criterion work in triangle congruence proofs?
3. What other angle pairs could be used in alternate congruence proofs?
4. Can we use any other congruence criteria like ASA or AAS here?
5. Why is the definition of segment bisector relevant in this proof?

Tip: In triangle proofs, always check if vertical or alternate interior angles can establish angle congruence.