If the complement of an angle is one-fourth of its supplementary angle, then the angle is

Options

A. `60^@`
B. `30^@`
C. `90^@`
D. `120^@`

Answer

To solve the problem step by step, we need to define the relationships between the angle, its complement, and its supplement.

### Step-by-Step Solution:

1. **Define the Angle**:
Let the angle be \( \theta \).

2. **Find the Complementary Angle**:
The complementary angle of \( \theta \) is given by:
\[ 90^\circ - \theta \]

3. **Find the Supplementary Angle**:
The supplementary angle of \( \theta \) is given by:
\[ 180^\circ - \theta \]

4. **Set Up the Equation**:
According to the problem, the complement of the angle is one-fourth of its supplementary angle. This can be expressed as:
\[ 90^\circ - \theta = \frac{1}{4}(180^\circ - \theta) \]

5. **Multiply Both Sides by 4**:
To eliminate the fraction, multiply both sides of the equation by 4:
\[ 4(90^\circ - \theta) = 180^\circ - \theta \]
This simplifies to:
\[ 360^\circ - 4\theta = 180^\circ - \theta \]

6. **Rearrange the Equation**:
Move all terms involving \( \theta \) to one side and constant terms to the other side:
\[ 360^\circ - 180^\circ = 4\theta - \theta \]
This simplifies to:
\[ 180^\circ = 3\theta \]

7. **Solve for \( \theta \)**:
Divide both sides by 3 to find \( \theta \):
\[ \theta = \frac{180^\circ}{3} = 60^\circ \]

### Final Answer:
The angle \( \theta \) is \( 60^\circ \).