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Answer :
Given:
Area (\(\triangle{DRC}\)) = Area (\(\triangle{DPC}\)).
As, \(\triangle{DRC}\) and \(\triangle{DPC}\) lie on the same base DC and have equal area, therefore, they must lie between the same parallel lines.
\(\because \) DC || RP
\(\therefore \) DCPR is a trapezium.
Also, Area (\(\triangle{BDP}\)) = Area (\(\triangle{ARC}\)).
\(\Rightarrow \) Area (\(\triangle{BDP}\)) - Area (\(\triangle{DPC}\)) = Area (\(\triangle{ARC}\)) - Area (\(\triangle{DRC}\))
\(\therefore \) Area (\(\triangle{BDC}\)) = Area(\(\triangle{ADC}\))
\(\because \) \(\triangle{BDC}\) and \(\triangle{ADC}\) lie on the same base CD and have equal areas, they must lie between the same parallel lines.
\(\therefore \) AB || CD
Hence, it is proved that ABCD is a trapezium.