Final answer:
To show that APQR and ASTU are congruent by ASA postulate, we need to identify a pair of congruent angles adjacent to the given congruent side PQ or ST. The correct answer is ZQ ≅ ZT, which are angles adjacent to the congruent sides, therefore option C is the correct answer.
Step-by-step explanation:
To demonstrate that quadrilaterals APQR and ASTU are congruent by the Angle-Side-Angle (ASA) postulate, we need to show two angles and the included side are congruent. Since it is given that sides PQ and ST are congruent (PQ ≅ ST) and angles ZPZS are congruent (ZP ≅ ZS), we need one more pair of congruent angles that include the side already known to be congruent (PQ or ST). The answer choices are angles R and U (A), angles P and Q (B), angles Q and T (C), and angles Q and U (D). For ASA, the angle that is needed to prove congruence must be adjacent to the given congruent side, hence we can eliminate choice B because those angles are not adjacent to the congruent sides. Among the remaining options, angle Q (ZQ) is at the end of side PQ while angle T (ZT) and angle U (ZU) are at the ends of side ST. Therefore, the correct answer would be the pair of angles that include the given side ST. Since ZT is adjacent to ST, the congruent angle pairs needed for ASA postulate are ZQ and ZT, making option C the correct answer.