Parallelograms - Same Base, Same Parallels
Consider two parallelograms ABCD and ABEF, on the same base AB, and between the same parallels, as shown below:
What will be the relation between the areas of these two parallelograms?
Theorem: Parallelograms on the same base and between the same parallels are equal in area.
Proof: Consider the figure presented above. Can you see that \(\Delta BCE\) and \(\Delta ADF\) will be congruent? This is easy to show. We have:
BC = AD (opposite sides of a parallelogram are equal)
\(\angle BCE\) = \(\angle ADF\) (corresponding angles)
\(\angle BEC\) = \(\angle AFD\) (corresponding angles)
By the ASA criterion, the two triangles are congruent, which means that their areas are equal. Now,
area(ABCD) = area(ABED) + area(\(\Delta BCE\))
area(ABEF) = area(ABED) + area(\(\Delta ADF\))
area(ABCD) = area(ABEF)
This completes the proof.
Next, consider a parallelogram ABCD and a rectangle ABEF on the same base and between the same parallels:
Clearly, their areas will be equal. Now, the length and height (width) of the rectangle have been marked as l and w respectively. Therefore,
area(ABCD) = area(ABEF) = l × w
This means that the area of any parallelogram is equal to the product of its base and its height (the height of a parallelogram can be defined as the distance between its base and the opposite parallel).
Now, consider the following figure, which shows a parallelogram ABCD and a triangle ABE on the same base AB and between the same parallels:
What will be the relation between the areas of these two figures? Let us complete the parallelogram ABEF, as shown below:
Now, we have:
area(\(\Delta ABE\)) = ½ area(ABFE)
= ½ area(ABCD)
Thus, the area of the triangle is exactly half of the area of the parallelogram. Let us define the height of a triangle as the distance between the base and the parallel through the opposite vertex. We can therefore say that the area of the triangle will be:
Area = ½ × base × height
This is shown below:
Note that any of the three sides of the triangle can be taken as the base, but then the height will change accordingly.