Parallelogram Area Ratio Theorem

The area of a parallelogram is calculated using the formula:

$ \text{Area} = \text{Base} \times \text{Height} $

Conditions for Equal Areas

Consider two parallelograms, Parallelogram 1 and Parallelogram 2.

• Equal Bases: The problem states that their bases are equal. Let Base$_1$ be the base of Parallelogram 1 and Base$_2$ be the base of Parallelogram 2. Then, $ \text{Base}_1 = \text{Base}_2 $
• Same Parallels: When two parallelograms are between the same parallels, they share the same perpendicular distance between these parallels. This distance is the height of both parallelograms. Let Height$_1$ be the height of Parallelogram 1 and Height$_2$ be the height of Parallelogram 2. Then, $ \text{Height}_1 = \text{Height}_2 $

Calculating the Area Ratio

Using the area formula:

• Area$_1 = \text{Base}_1 \times \text{Height}_1$
• Area$_2 = \text{Base}_2 \times \text{Height}_2$

Since $ \text{Base}_1 = \text{Base}_2 $ and $ \text{Height}_1 = \text{Height}_2 $, it follows that:

$ \text{Area}_1 = \text{Area}_2 $

The ratio of their areas is:

$ \frac{\text{Area}_1}{\text{Area}_2} = \frac{\text{Area}_1}{\text{Area}_1} = \frac{1}{1} $

Therefore, the ratio of the areas is $ 1 : 1 $.