The angles of a quadrilateral are in the ratio `3 : 5 : 7 : 9`. What is the difference between the least and the greatest angles of the quadrilateral?

Options

A. `90^(@)`
B. `50^(@)`
C. `60^(@)`
D. `72^(@)`

Answer

To solve the problem of finding the difference between the least and the greatest angles of a quadrilateral whose angles are in the ratio 3:5:7:9, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Angles**:
Let the angles of the quadrilateral be represented as:
- Angle 1 = 3x
- Angle 2 = 5x
- Angle 3 = 7x
- Angle 4 = 9x

2. **Use the Sum of Angles in a Quadrilateral**:
We know that the sum of the angles in a quadrilateral is 360 degrees. Therefore, we can set up the equation:
\[ 3x + 5x + 7x + 9x = 360 \]

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ (3 + 5 + 7 + 9)x = 360 \]
This simplifies to:
\[ 24x = 360 \]

4. **Solve for x**:
To find the value of x, divide both sides of the equation by 24:
\[ x = \frac{360}{24} = 15 \]

5. **Calculate Each Angle**:
Now, substitute the value of x back into the expressions for each angle:
- Angle 1 = 3x = 3(15) = 45 degrees
- Angle 2 = 5x = 5(15) = 75 degrees
- Angle 3 = 7x = 7(15) = 105 degrees
- Angle 4 = 9x = 9(15) = 135 degrees

6. **Identify the Least and Greatest Angles**:
From the calculated angles:
- Least angle = 45 degrees
- Greatest angle = 135 degrees

7. **Find the Difference**:
Finally, calculate the difference between the greatest and least angles:
\[ \text{Difference} = 135 - 45 = 90 \text{ degrees} \]

### Final Answer:
The difference between the least and the greatest angles of the quadrilateral is **90 degrees**.

---