Question

ABCD is a parallelogram. BM bisects angle ABC and
DN bisects angle ADC . Prove that BNDM is a parallelogram and BM = DN.

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Answer 1

Answer: ↓

Step-by-step explanation:

∵ABCD is a parallelogram(given)

∴∠ABC=∠CDA(opp. ∠s of //gram)

∵BM bisects angle ABC and  DN bisects angle ADC.

∴∠ABM=∠MBC=∠CDN=∠NDA

∵∠MBN=∠NDM(opp. ∠s equal)

∴BNDM is a parallelogram.

∴BM = DN(opp. sides of //gram)

Answer 2

Step-by-step explanation:

∠B=∠D

1/2∠B=1/2∠D

∠ABM= ∠CDN ...(1)

In ΔABM and ΔCDN

∠ABM= ∠CDN (FROM 1)

AB = CD (Opp. sides in a parallelogram are equal)

∠A=∠C   (Opp. angles in a parallelogram are equal)

⇒ΔABM  ≅ ΔCDN (ASA Congruence rule)

⇒BM=DN (BY CPCT)

Also, by CPCT  BM║DN

So, BMDN is a Parallelogram (a quadrilateral with any two sides equal and    

                                                   parallel is a parallelogram)          

Hence proved that BM=DN and BMDN is a parallelogram.

I hope it will help you. :)