Two parallelograms stand on equal bases and between the same parallels. The ratio of their areas is :

Options

A. `1:1`
B. `sqrt(2):1`
C. `1:3`
D. `1:2`

Answer

To find the ratio of the areas of two parallelograms that stand on equal bases and between the same parallels, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of Parallelograms**:
- The area of a parallelogram is given by the formula:
\[ \text{Area} = \text{Base} \times \text{Height} \]
- In this case, both parallelograms share the same base and are situated between the same parallels.

2. **Identify the Bases and Heights**:
- Let the base of both parallelograms be denoted as \( AB \).
- Let the heights from the base \( AB \) to the opposite sides of the parallelograms be denoted as \( GH \) for both parallelograms.

3. **Calculate the Area of Each Parallelogram**:
- For the first parallelogram \( ABCD \):
\[ \text{Area}_{ABCD} = AB \times GH \]
- For the second parallelogram \( ABEF \):
\[ \text{Area}_{ABEF} = AB \times GH \]

4. **Set Up the Ratio of the Areas**:
- The ratio of the areas of the two parallelograms can be expressed as:
\[ \text{Ratio} = \frac{\text{Area}_{ABCD}}{\text{Area}_{ABEF}} = \frac{AB \times GH}{AB \times GH} \]

5. **Simplify the Ratio**:
- Since both areas are equal, the ratio simplifies to:
\[ \text{Ratio} = \frac{1}{1} = 1 \]

6. **Conclusion**:
- Therefore, the ratio of the areas of the two parallelograms is:
\[ 1 : 1 \]

### Final Answer:
The ratio of their areas is \( 1 : 1 \).