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प्रश्न
In the given figure, AP is the bisector of ∠A and CQ is the bisector of ∠C of parallelogram ABCD. 
Prove that APCQ is a parallelogram.
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उत्तर
Construction: Join AC
Proof:
∠BAP = `1/2`∠A ...( AP is the bisector of ∠A )
∠DCQ = `1/2`∠C ...( CQ is the bisector of ∠C )
⇒ ∠BAP = ∠DCQ ....(i)....[ ∠A = ∠R ( Opposite angles of a parallelogram.) ]
Now,
∠BAC = ∠DCA ....(ii)....[ Alternate angles since AB || DC ]
Subtracting (ii) from (i), We get
∠BAP - ∠BAC = ∠DCQ - ∠DCA
⇒ ∠CAP = ∠ACQ
⇒ AP || QC .....( Alternate angles are equal )
Similarly, PC || AQ.
Hence, APCQ is a parallelogram.
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