Question

9) Prove that a quadrilateral is a parallelogram if a pair of opposite sides is equal and
parallel.
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Answer 1

Answer:

Given: ABCD is quadrilateral and AB║CD, AB=CD.

To prove: ABCD is a parallelogram

Proof: AC is a transversal and also AB║CD, therefore

∠BAC=∠DCA(Alternate angles)

In ΔADC and ΔCBA, we have

AB=CD(Given)

∠BAC=∠DCA(Alternate angles)

AC=CA(Common)

ΔADC≅ΔCBA by the SAS rule.

Hence, by CPCT, DA=BC

Thus, Both the pair of opposite sides are equal in the quadrilateral ABCD, therefore ABCD is a parallelogram.

Hence proved.

Answer 2

Step-by-step explanation:

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 Read more on Sarthaks.com - https://www.sarthaks.com/75238/quadrilateral-is-parallelogram-if-pair-of-opposite-sides-is-equal-and-parallel-prove-that