1.

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

Answer»

ong>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.

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