Question

In the adjoining figure ,ABCD is a parallelogram . If P and Q are points on AD and BC respectively such that AP=1/3 AD and CQ = 1/3 BC,Prove that AQCP is a Parallelogram.

Answer 1

Answer:

We know that the opposite sides of a parallelogram are equal

Step-by-step explanation:

Consider △ ABQ and △ CDP

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 AQCP is a parallelogram.