The parallelogram on same base and between the same parallel have the same area

True

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False

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Solution

The correct option is A True

Here, two parallelograms $A B C D$ and $E F C D,$ that have the same base $C D$ and lie between same parallels $A F$ and $C D$ .
Since opposite sides of parallelogram are parallel
$A D ∥ B C$ with transversal $A B$
$\Rightarrow$ $∠ D A B = ∠ C B F$ [ Corresponding angles ] ----- ( 1 )
$E D ∥ F C$ with transversal $E F$
$\Rightarrow$ $∠ D E A = ∠ C F E$ [ Corresponding angles ] ----- ( 2 )
In $△ A E D$ and $△ B F C$
$\Rightarrow$ $∠ D E A = ∠ C F E$ [ From ( 2 ) ]
$\Rightarrow$ $∠ D A B = ∠ C B F$ [ From ( 1 ) ]
$\Rightarrow$ $A D = B C$ [ Opposite sides of parallelogram are equal ]
$∴$ $△ A E D ≅ △ B F C$ [ AAS congruence property ]
$\Rightarrow$ $a r (△ A E D) = a r (△ B F C)$ [ Area of congruent figures are equal ]
Adding $a r (E B C D)$ on both sides,
$\Rightarrow$ $a r (△ A E D) + a r (E B C D) = a r (△ B F C) + a r (E B C D)$
$\Rightarrow$ $a r (A B C D) = a r (E F C D)$
$∴$ The parallelogram based upon the same base and between the same parallel lines are $e q u a l i n a r e a$

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Similar questions

Choose the correct statements from the following:

Statement 1: All the parallelograms having the same base and lying between the same parallels have equal areas.

Statement 2: All the parallelograms having the same base and lying between the same parallels have same perimeters.

Statement 3: Of all the parallelograms having same base and lying between same parallels, rectangle will have the smallest perimeter.

Statement 4: All the parallelograms having equal areas must have the same base but they may or may not lie between the same parallels.

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