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Properties of a Parallelogram - The Diagonals of a Parallelogram Bisect Each Other.

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theorem

Theorem : The diagonals of a parallelogram bisect each other.   
proof : 

Since ABCD is a parallelogram 
`therefore ` AB || DC and AD || BC
Now, AB parallel to DC and AC is the transversal intersecting them at A and C respectively. 
∠BAC =∠ DCA (Alternate interior angles)
Thus, ∠ BAO = ∠ DCO ... (1)
As , AB || DC and BD intersect them at B and D respectively. 
∠ ABD= ∠CDB (Alternate interior angles) 
So, ∠ ABD= ∠CDO   ...(2) 
Now , In triangle AOB and Triangle COD , we have 
 ∠BAO = ∠  DCO [From (1)] 
AB = CD ( Opposites sides of a parallelogram )
∠ ABO =∠ CDO  [From (2)] 
`therefore`  ∆ AOB ≅ ∆ COD   (by ASA congruence criterion)
Thus , OA = OC and OB = OD (C.P.C.T)

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