PQRS is a parallelogram in which PQ is produced to T such that QT=PQ. prove that ST bisects RQ.


We have PQRS is a parallelogram.

PQ = SR  [opposite sides of parallelogram are equal]

But PQ = QT  [given] 

so, SR = QT 

Since SR || PT and ST is a transversal

∠RSM = ∠QTM  [Alternate interior angles]

In triangle SMR and TMQ

∠SMR = ∠TMQ  [alternate interior angles]

∠RSM = ∠QTM  [Alternate interior angles]

SR = QT  [proved above]

ΔSMR is congruent to ΔTMQ  [AAS CRITERIA]

⇒ RM = QM  [CPCT]

so, ST bisects RQ.

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