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In Figure, a r\ ( D R C)=a r( D P C)a n ...

In Figure, `a r\ ( D R C)=a r( D P C)a n d\ a r( B D P)=a r( A R C)dot` Show that both the quadric-laterals `A B C D\ a n d\ D C P R` are trapeziums.

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We have to prove quadrilaterals `ABCD` and `DCPR` are trapeziums.
It is given that `Area (triangleDRC) = Area (triangleDPC)`
As `triangleDRC` and `triangleDPC` are lying on the same base `DC` having equal areas, therefore, they must lie between the same parallel lines.
According to Theorem 9.3:
Two triangles having the same base (or equal bases) and equal areas lie between the same parallels.
`DC || RP`
Therefore, `DCPR` is a trapezium.
It is also given that `Area (triangleBDP) = Area (triangleARC)`
Now, subtract `ar (triangleDPC)` fromm `ar (triangleBDP)` and `ar (triangleDRC)` from `ar (triangleARC)`
`ar (triangleBDP) - ar (triangleDPC) = ar (triangleARC) - ar (triangleDRC)` [Since, `ar (triangleDPC) = ar (triangleDRC`)]`ar (triangleBDC) = ar (triangleADC)`
Since `triangleBDC` and `triangleADC` are on the same base `CD` having equal areas, they must lie between the same parallel lines. (According to Theorem 9.3)
`AB || CD`
Therefore, `ABCD` is a trapezium.
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