1. **Problem Statement:** Given triangle PSR with point S on base QR, and PS \cong QS, angles \angle P = 48^\circ and \angle S = 84^\circ. We need to determine which comparisons among PQ, QR, PR, QS, and SR are true. 2. **Key Information and Rules:** - PS \cong QS means segments PS and QS are equal. - The sum of angles in triangle PSR is 180^\circ. - The triangle inequality states that in any triangle, the side opposite the larger angle is longer. 3. **Find the missing angle:** $$\angle R = 180^\circ - 48^\circ - 84^\circ = 48^\circ$$ 4. **Analyze triangle PSR:** - Angles: \angle P = 48^\circ, \angle S = 84^\circ, \angle R = 48^\circ. - Since \angle S is largest, side PR opposite \angle S is the longest side. 5. **Analyze triangle PQS:** - Since PS \cong QS, triangle PQS is isosceles with PS = QS. 6. **Compare sides:** - PR is longest side in triangle PSR. - PQ is a side adjacent to S and P. - QR is base containing points Q and R. 7. **Evaluate each comparison:** - PQ > QR? No, because QR includes QS and SR, and QS = PS. - PQ > PR? No, PR is longest side. - PQ < PR? Yes, since PR is opposite largest angle. - QS \cong SR? No, only PS \cong QS given. - SR < PS? Yes, since PS = QS and SR is part of QR excluding QS. **Final answers:** - PQ < PR - SR < PS