Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).

Solution:

△DAC and △DBC lie on the same base DC and between the same parallels AB and CD.

Ar(△DAC) = ar(△DBC)

⇒   ar(△DAC) – ar(△DOC) = ar(△DBC) – ar(△DOC)

⇒  ar(△AOD) = ar(△BOC)