Question

Prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Answer 1

selected Jun 14, 2018 by Golu   Given: A quadrilateral ABCD whose diagonals AC and BD bisect each other at point O. i.e., OA = OC and OB = OD  To prove: ABCD is a parallelogram  i.e., AB ║ DC and AD ║ BC.  Proof: In ∆AOB and ∆COD  OA = OC [Given]  OB = OD [Given}  And, ∠AOB = ∠COD [Vertically opposite angles]  ⇒ ∆AOB ≅ ∆COD [By SAS]  ⇒ ∠1 = ∠2 [By cpctc]  But these are alternate angles and whenever alternate angles are equal, the lines are parallel.  ∴ AB is parallel to DC i.e., AB ║ DC  Similarly,  ∆AOD ≅ ∆COB [By SAS]  ⇒ ∠3 = ∠4  But these are also alternate angles ⇒ AD ║ BC  AB ║ DC and AD ║ BC ⇒ ABCD is a parallelogram.