P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.

Explanation

Given ABCD is a parallelogram whose diagonals bisect each other at O.
To show PQ is bisected at O.

In `Delta`ODP and `Delta`OBQ,
`" "angleBOQ=anglePOD" "` [since, vertically opposite angles]
`" "angleOBQ=angleODP" "`[alternate interior angles]
and `" "OB=OD" "[given]`
`therefore" "DeltaODP~=DeltaOBQ" "`[by ASA congruence rule ]
`therefore" "OP=OQ" "`[by CPCT rule]
So, PQ is bisected at O. `" "` Hence proved.