Question

40. Prove that a quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

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Answer 1

Step-by-step explanation:

Theorem :-

A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

Proof :-

Given: A quadrilateral ABCD in which AB ║ DC and AB = DC.

To Prove: ABCD is a parallelogram

i.e., AB ║ DC and AD ║ BC.

Construction: Join A and C.

Proof : Since AB is parallel to DC and AC is transversal

∠BAC = ∠DCA [Alternate angles]

AB = DC [Given]

And AC = AC [Common side]

⇒ ∆BAC ≅ ∆DCA [By SAS]

⇒ ∠BCA = ∠DAC [By cpctc]

But these are alternate angles and whenever alternate angles are equal, the lines are parallel.

⇒ AD ║ BC

Now, AB ║ DC (given) and AD ║ BC [Proved above]

⇒ ABCD is a parallelogram