Question

if the diagonals of a quadilateral bisect each other , quadilateral is a parallelogram

Answer 1

Hey there, 

ABCD is a ⇵gm.
AB⇵CD.
∠ABD=∠BDC since they are alternate
Draw XY ⇵BC
∠XOB=∠DOY since they are vertically opposite
ΔBXO≡ΔDYO by ASA criterion
So OX=OY and converse.

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Answer 2

Answer:

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. 

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