From the figure we know that △ AOD and △ DCB lie on the same base CD and between two parallel lines DC and AB.
We know that the triangles lying on the same base and parallels have equal area.
Consider △ CDA and △ CDB
It can be written as
Area of △ CDA = Area of △ CDB
So we get
Area of △ AOD = Area of △ ADC – Area of △ OCD
In the same way
Area of △ BOC = Area of △ CDB – Area of △ OCD
So we get
Area of △ BOC = Area of △ ADC – Area of △ OCD
Area of △ AOD = Area of △ BOC
Therefore, it is proved that ar (△ AOD) = ar (△ BOC).