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In the trapezium AB // DC,
So ar(ABD) = ar(ABC) {on same base and between same parallels}
= ar(ABD) - ar(AOB) = ar(ABC) - ar(AOB
= ar(AOD) = ar(BOC)

Hope that helps !!
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In trapezium ABCD , as triangle DAB and triangle CAB lie on same base AB and between same parallels AB parallel to CD , so area ADB equals to area ACB
subtract area AOB from b I th sides, area ADB - AOB = area CAB -AOB          
So area AOD = area COB
hence proved

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