Question

if the diagonal of a Quadrilateral bisects each other the n prove that it is a parallelogram
​

Answer 1

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.

Answer 2

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.

please mark the answer as brainliest and thank the answer.