Prove: if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. abcd is a parallelogram.

Answer with explanation:

 A Quadrilateral A B CD, in which diagonals intersect each other at O.

  In Δ AOB and ΔCOD

AO=OC-----[Given]

∠AOB=∠COD→→→Vertically Opposite Angles

BO=OD------[Given]

Δ AOB ≅ ΔCOD-------[SAS]

1.⇒AB=CD→→→C PCT

Now, In ΔA OD and Δ B O C

 A O=O C-----[Given]

 BO=OD-----[Given]

∠AOD=∠BOC→→→Vertically Opposite angles

ΔA OD ≅ Δ B O C------[S AS]

2.⇒  AD=BC→→[C P CT]

From 1 and 2,

If in a Quadrilateral , opposite sides are equal, then it is a Parallelogram.

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