Consider the below parallelogram ABCD with diagonals AC and BD.
Given:
We know that opposite sides of parallelogram are equal.
AB = CD
AD = BC
Also in parallelogram, opposite sides are parallel to each other.
AB || CD
AD || BC
To prove:
Diagonals bisect each other.
AO = OC
DO = OB
Proof:
Consider triangle AOD and BOC.
AD = BC (given)
∠DAO = ∠BCO (alternate interior angle)
∠ADO = ∠OBC (alternate interior angle)
By ASA congruency condition, both the triangles are congruent.
\mathtt{\triangle AOD\cong \triangle BOC}
By rule of congruency, we can say that AO = OC and DO = OB.
Hence in parallelogram, the diagonals bisect each other.