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. `50^@`
B. `60^@`
C. `72^@`
D. `90^@`

Answer

To solve the problem, we need to find the angles of the quadrilateral given in the ratio 3:5:7:9 and then determine the difference between the least and the greatest angles.

### Step-by-Step Solution:

1. **Understand the Ratios**: The angles of the quadrilateral are given in the ratio 3:5:7:9. Let's denote the angles as:
- Angle A = 3x
- Angle B = 5x
- Angle C = 7x
- Angle D = 9x

2. **Sum of Angles in a Quadrilateral**: The sum of the angles in any quadrilateral is 360 degrees. Therefore, we can write the equation:
\[ 3x + 5x + 7x + 9x = 360 \]

3. **Combine Like Terms**: Combine the terms on the left side:
\[ 24x = 360 \]

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

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

6. **Identify the Least and Greatest Angles**: From the calculated angles:
- Least angle = 45 degrees (Angle A)
- Greatest angle = 135 degrees (Angle D)

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

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

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