To determine if triangle PQR is congruent to triangle TSR by the SAS (Side-Angle-Side) Congruence Theorem, we need two sides and the included angle to be congruent between the two triangles.
Given that point R is the midpoint of both QS and PT, we have:
- \( QR = SR \) (since R is the midpoint of QS)
- \( PR = RT \) (since R is the midpoint of PT)
This means we can establish one side congruence between the two triangles as \( QR \cong SR \) and \( PR \cong RT \).
Now, for the SAS criteria, we need one angle to be congruent.
Among the options provided:
A. QR is congruent to SR - This would confirm one pair of sides. B. Angle PRQ is congruent to angle TRS - This states that the angle between the sides PR and QR is congruent to the angle between the sides TR and SR, which satisfies the angle condition for SAS. C. PQ is congruent to TS - This would provide another side but doesn't confirm an included angle. D. Angle QPR is congruent to angle STR - Similar to option B, this could also work but doesn't align with the sides we already know.
In conclusion, the most relevant option that provides the necessary information to confirm the included angle for SAS is:
B. Angle PRQ is congruent to angle TRS.