Final answer:
To prove PQ ≅ SR, we first establish that ΔQPS and ΔRSP are congruent using the SAA condition as PS is the common side and ∠Q ≅ ∠R and ∠QPS ≅ ∠RSP. Then by CPCTC, corresponding sides PQ and SR are congruent.
Step-by-step explanation:
To prove that PQ ≅ SR using CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we need to show that the triangles are congruent first. Since it's given that ∠QPS ≅ ∠RSP and ∠Q ≅ ∠R, we look for a third pair of congruent parts to prove the congruence of the triangles.
If triangles QPS and RSP share a side, namely PS, then we have the Side-Angle-Angle (SAA) congruence condition (which is equivalent to the Angle-Side-Angle, or ASA, condition), because the shared side (PS) acts as the included side between the two congruent angles. Thus, ΔQPS ≅ ΔRSP by SAA.
Finally, by CPCTC, since the two triangles are congruent, their corresponding parts are congruent as well. Therefore, segment PQ ≅ SR as these are corresponding sides of the congruent triangles.