Question

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.

Answer 1

Construction: Join AC

Proof:
∠BAP = ${\displaystyle \frac{1}{2}}$∠A             ...( AP is the bisector  of ∠A )

∠DCQ = ${\displaystyle \frac{1}{2}}$∠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.