Math - history - geometry

Suppose "parallelogram" has just been defined as a quadrilateral with two pairs of parallel sides and the following theorem is the first property of parallelograms to be deduced. There is something wrong with the proof of the following theorem. Find the error and correct the proof. Theorem: The diagonals of a parallelogram bisect each other. "Proof": Let ABCD be a parallelogram with diagonals AC' and BD intersecting at O. AD is parallel to BC and AB is parallel to DC since ABCD is given as a parallelogram. Since alternate interior angles formed by parallel lines are congruent, ZCAD ~ ZACB and LADB ~ LDBC. AD ~ BC, since opposite sides of a parallelogram are equal in length. Thus AAOD ~ ACOB by ASA congruence. Thus AO ~ OC and BO ~ OD, since corresponding parts of congruent triangles are congruent