Question

prove that " a quadrilateral is a parrallelogram if a pair of opposite sides is equal and parallel "​

Answer 1

Answer: 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