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)