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Proposition:

Let the diagonals AC and BD of the parallelogram ABCD intersect at O. It is required to prove that AO = CO, BO = DO.


Proof:

Step-1: The lines AB and DC are parallel and AC is their transversal.
Therefore, ∠BAC = alternate ∠ACD. (Alternate angles are equal)

Step-2: The lines BC and AD are parallel and BD is their transversal
Therefore, ∠BDC = alternate ∠ABD. (Alternate angles are equal)

Step-3: Now, between ΔAOB and ΔCOD
∠OAB = ∠OCD, ∠OBA = ∠ODC and AB = DC .
So ΔAOB ≅ ΔCOD.
Therefore, AO = CO and BO = DO. (Proved)

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