Question

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

parallelogram.​

Answer 1

Answer:

Step-by-step explanation:

ABCD is an quadrilateral with AC and BD are diagonals intersecting at O.

It is given that diagonals bisect each other.

∴  OA=OC and OB=OD

In △AOD and △COB

⇒  OA=OC                                [ Given ]

⇒  ∠AOD=∠COB                  [ Vertically opposite angles ]

⇒  OD=OB                               [ Given ]

⇒  △AOD≅△COB                 [ By SAS Congruence rule ]

∴  ∠OAD=∠OCB         [ CPCT ]  ----- ( 1 )

Similarly, we can prove  

⇒  △AOB≅△COD

⇒  ∠ABO=∠CDO       [ CPCT ]     ---- ( 2 )

For lines AB and CD with transversal BD,

⇒  ∠ABO and ∠CDO are alternate angles and are equal.

∴  Lines are parallel i.e. AB∥CD

For lines AD and BC, with transversal AC,

⇒  ∠OAD and △OCB are alternate angles and are equal.

∴  Lines are parallel i.e. AD∥BC

Thus, in ABCD, both pairs of opposite sides are parallel.

∴  ABCD is a parallelogram.