VerifiedHint: For the above question first of all we will have to suppose two parallelograms lying on the same base and between the same parallels. Then we will make the triangle congruent and use the property that congruent figures have the same area.
Complete step-by-step solution
We have been asked to prove that parallelograms on the same base and between the same parallels are equal in area.
Let us suppose ABCD and EFCD are two parallelograms lying on the same base CD and between the same parallels AF and CD.
We have to prove that area of parallelograms ABCD and EFCD are equal.
In $\Delta ADE\ and\ \Delta BCF$
$\angle DAE=\angle CBF$
Since, AD is parallel to BC and AB is a transversal. So these are corresponding angles.
$\angle AED=\angle BFC$
Since, DE is parallel to CF and EF is a transversal. So these are corresponding angles.
$AD=BC$
Since, ABCD is a parallelogram and these are opposite sides of a parallelogram.
Hence, $\Delta ADE\cong \Delta BCF$, by ASA congruence rule i.e., Angle – Side – Angle congruence rule. We know that congruent figures have the same area.
$\Rightarrow area\left( \Delta ADE \right)=area\left( \Delta BCF \right)..........\left( 1 \right)$
Now, we have:
$area\left( ABCD \right)=area\left( ADE \right)\bot area\left( EDCB \right)$
Since, $area\left( \Delta ADE \right)=area\left( \Delta BCF \right)$ from (1)
$\begin{align}
& \Rightarrow area\left( ABCD \right)=area\left( \Delta BCF \right)+area\left( EDCB \right) \\
& =area\left( EDCF \right) \\
\end{align}$
Thus, the area of the two parallelograms ABCD and EFCD are the same.
Hence, it is proved that parallelograms on the same base and between the same parallels are equal in area.
Note: Remember the point in these types of question diagrams is very important. So, first of all draw the diagram according to the question.Also, you can remember this statement which we have proved as it will help you a lot to solve the questions related to parallelogram.
