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If the opposite sides of a quadrilateral are equal and parallel then prove that it is a parallelogram

Posted by A K 5 years, 3 months ago

CBSE > Class 09 > Mathematics

1 answers

Gaurav Seth 5 years, 1 month ago

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

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CBSE > Class 09 > Mathematics