Question

In fig, diagonal AC and BD of quadrilateral ABCD intersect at O such that OB=OD. If AB=CD, then show that :
1) ar(DOC) = ar(AOB)
2) ar(DCB) = ar(ACB)
3) DA || CB or ABCD is a parallelogram.
plz solve this..

Answer 1

1)
In triangle DOC and AOB
DO=OB (given)
AB=CD (given)
angle DOC = angle AOB (verticaly opp angle)
=> triangle DOC =~ triangle AOB
ar (DOC) = ar (AOB) by cpct

2)
In triangle DCB and ACB
AB=CD (given)
CB = CB (common)
angle CDB = angle CAB (angle in same segment)
=> triangle DCB =~ triangle ACB
ar DCB = ar ACB by cpct