1. Given: \( \angle P \cong \angle S \); \( O \) is the midpoint of \( \overline{P S} \).
2. Definition of congruent angles.
3. \( \angle P \cong \angle S \) implies \( \angle S \cong \angle P \).
4. \( \angle S \cong \angle P \) implies \( \angle S \) and \( \angle P \) are corresponding angles.
5. Corresponding angles are congruent.
6. \( \overline{P S} \) is a line segment.
7. \( O \) is the midpoint of \( \overline{P S} \).
8. Definition of midpoint.
9. \( O \) divides \( \overline{P S} \) into two congruent segments.
10. \( R \) and \( Q \) are points on \( \overline{P S} \).
11. \( O \) divides \( \overline{P S} \) into two congruent segments, so \( O \) is the midpoint of \( \overline{R Q} \).
12. Definition of midpoint. Happy to help