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**Hint:**

We will first find the area of the parallelogram using the formula for the area of a parallelogram. Then we will find the area of the rectangle using the formula for the area of a rectangle. We will compare both the areas to prove that both the areas are equal.

**Formulas used:**We will use the following formulas:

1) The area of a rectangle \[{A_r} = l \times b\], where \[l\] is the length and \[b\] is the base.

2) The area of a parallelogram, \[{A_p} = b \times h\] , where \[h\] is the height and \[b\] is the base.

**Complete step by step solution:**

We will first prove the \[{\rm{ar}}\left( {ABCD} \right) = DC \times AL\].

(ii) We will find the area of parallelogram ABCD. We know that the area of a parallelogram is the product of the length of its height and base. So, we can write

\[ar\left( {ABCD} \right) = b \times h\]

The height of a parallelogram is the perpendicular distance between its 2 parallel opposite sides and the base of the parallelogram is its lowermost side. We can see from the figure that the height of the parallelogram is \[AL\] and the base is \[DC\]. We will substitute these values in the formula for the area of a parallelogram:

\[ \Rightarrow ar\left( {ABCD} \right) = AL \times DC\]

Hence, we have proved \[{\rm{ar}}\left( {ABCD} \right) = DC \times AL\].

Now, we will prove the \[{\rm{ar}}\left( {ABCD} \right) = {\rm{ar}}\left( {EFCD} \right)\].

We will find the area of the rectangle EFCD. We know that the area of a rectangle is its length times breadth.

\[ \Rightarrow ar\left( {EFCD} \right) = {\rm{length}} \times {\rm{breadth}}\]

Usually, the longer side of a rectangle is taken as its length and the shorter side is taken as its breadth. The breadth of a rectangle is always perpendicular to its length just like the height of a parallelogram. We can see from the figure that the length of the rectangle is \[EF\] and the breadth is \[ED\]. We will substitute these values in the formula for the area of a rectangle.

Therefore,

\[ \Rightarrow ar\left( {EFCD} \right) = EF \times ED\]

We know that the opposite sides of a rectangle are equal in length. So,

\[EF = DC\]

We can see from the figure that the breadth of the rectangle and the height of the parallelogram are equal. So, we can write

\[ED = AL\]

So, the area of rectangle EFCD can be rewritten as:

\[ \Rightarrow ar\left( {EFCD} \right) = DC \times AL\]

We know that the area of parallelogram ABCD is also \[AL \times DC\].

**$\therefore ar\left( ABCD \right)=ar\left( EFCD \right)$**

**Note:**

A parallelogram is a quadrilateral whose opposite sides are parallel and equal. A rectangle is a special kind of parallelogram with all its angles as right angles. The area of a rectangle is also the same as the area of a parallelogram as the base of the parallelogram is the rectangle’s length and the height of the parallelogram is the rectangle’s breadth.

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