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Properties of a Parallelogram - Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram

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theorem

Theorem : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
proof :  

In ∆ AOD and ∆ COB  (Given)
OD = OB   (Given) 
∠AOB = ∠COD (Vertically opposite angles are equal)
Therefore , ∆AOD ≅ ∆COB  (By SAS criterion of congruence)
So, ∠OAD = ∠OCB  ...(1) (C.P.C.T)
Now , lines AC intersects AD and BC at A and C respectively,
such that ∠OAD = ∠OCB  ...[from(1)]
`therefore` ∠OAD and ∠ OCB form a pair of alternate interior angles are equal.
Thus, AD || BC
Similarly , we can prove that AB || DC 
Hence , quadrilatera; ABCD is a parallelogram.

Shaalaa.com | Theorem : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

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