Question

If the diagonals of a quadrilateral bisect each other then prove that the quadrilateral is a parallelogram.

Answer 1

here is ur answer......

Answer 2

Answer:

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}  

∠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.