Question

The diagonal of quadrilateral bisect each other . Prove that the given quadrilateral is a parallelogram

Answer 1

THEOREM: If a quadrilateral has diagonals which bisect each other, then it is a parallelogram. THEOREM: If a quadrilateral has one set of opposite sides which are both congruent and parallel, then it is a parallelogram. This last method can save time and energy when working a proof!

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.

Hope it helps you