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

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A Diagram would be helpful

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A Diagram would be helpful

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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.

EDIT:

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.

∴

Thus proved.