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Question
In the given figure, MP is the bisector of ∠P and RN is the bisector of ∠R of parallelogram PQRS. Prove that PMRN is a parallelogram.
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Solution
Construction: Join PR.
Proof:
∠QPM = `(1)/(2)∠"P"` ....(AP is the bisector of ∠P)
∠SRN = `(1)/(2)∠"R"` ....(RN is the bisector of ∠R)
⇒ ∠QPM = ∠SRN (i)....[∠P =∠R (Opposite angles of a parallelogram)]
Now, ∠QPR = ∠SRN (ii)....[Alternate angles since PQ || SR)
Subtracting (ii) from (i), we get
∠QPM - ∠QPR = ∠SRN - ∠SRP
⇒ ∠RPM =∠PRN
⇒ PM || RN ...(Alternate angles are equal)
Similarly, RM || PN
Hence, PMPN is a parallelogram.
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