Prove that the diagonals of a parallelogram bisect each other.

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Solution

Let ABCD be the parallelogram. Therefore, AB || DC and AD||BC.

Consider triangle AOD and COB.

AD = BC (opposite sides of a parallelogram)

∠DAO = ∠BCO (Alterante angles)

∠ADO = ∠CBO (Alternate angles)

Therefore, by ASA congruency, the triangle are congruent.

Now AO = OC and BO = OD because they are corresponding sides of two congruent triangle. Thus, the diagonals of a parallelogram bisect each other.

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