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.

Answer 1

Construction: Join PR.

Proof:

∠QPM = ${\displaystyle \frac{1}{2}\angle \text{P}}$    ....(AP is the bisector of ∠P)

∠SRN = ${\displaystyle \frac{1}{2}\angle \text{R}}$     ....(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.