Question

ABCD is a parallelogram and points P and Q are the points on sides AD and BC respectively such that AP= 1/4 AD and CQ = 1/4 BC . Prove that BPDQ is parallelogram.​

Answer 1

Step-by-step explanation:

Consider △ ABQ and △ CDP

We know that the opposite sides of a parallelogram are equal AB = CD

So we get ∠ B = ∠ D

We know that DP = AD – PA

i.e. DP = 2/3 AD BQ = BC – CQ

i.e. BQ = BC – 1/3 BC BQ = (3-1)/3 BC

We know that AD = BC

So we get BQ = 2/3 BC = 2/3 AD

We get BQ = DP

By SAS congruence criterion

△ ABQ ≅ △ CDP

AQ = CP (c. p. c. t)

We know that PA = 1/3 AD

We know that AD = BC CQ = 1/3

BC = 1/3 AD

So we get PA = CQ

∠ QAB = ∠ PCD (c. p. c. t)… (1)

We know that ∠ QAP = ∠ A – ∠ QAB

Consider equation (1) ∠ A = ∠ C

∠ QAP = ∠ C – ∠ PCD

From the figure we know that the alternate interior angles are equal ∠ QAP = ∠ PCQ

So we know that AQ and CP are two parallel lines.

Therefore, it is proved that PAQC is a parallelogram.