Question

P, Q, R, S are four consecutive points on a circle such that PQ = RS. Prove that PR = QS

Answer 1

I am assuming you already have the diagram there, so here's the answer.
EDIT: This method only works if PQ is parallel to RS, otherwise it is in possible for PS and QR to be equal.

Construct radii OP, OQ, OR and OS

Prove ΔPOQ and ΔSOR congruent by SSS test.
∴ ∠POQ = ∠SOR (CPCT)
=> 180°-∠POQ = 180°-∠SOR
=> ∠QOR = ∠POS (these angles are supplementary to the last two, since they are linear pairs)

From here you can prove ΔQOR and ΔPOS congruent by SAS test.
∴ PS = QR (CPCT)

Thus proved.