The measure of an angle for which the measure of the supplement is four times the measure of the complement is

Options

A. `75^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the measure of an angle \( x \) such that its supplement is four times its complement. Let's go through the solution step by step.

### Step 1: Define the angle
Let the angle be \( x \).

### Step 2: Write expressions for the supplement and complement
- The supplement of an angle \( x \) is given by:
\[ \text{Supplement} = 180^\circ - x \]
- The complement of an angle \( x \) is given by:
\[ \text{Complement} = 90^\circ - x \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the supplement is four times the complement. We can write this as:
\[ 180^\circ - x = 4(90^\circ - x) \]

### Step 4: Expand the equation
Now, let's expand the right side of the equation:
\[ 180^\circ - x = 360^\circ - 4x \]

### Step 5: Rearrange the equation
To isolate \( x \), we can add \( 4x \) to both sides and subtract \( 180^\circ \) from both sides:
\[ 180^\circ - x + 4x = 360^\circ \]
\[ 180^\circ + 3x = 360^\circ \]

### Step 6: Solve for \( x \)
Now, subtract \( 180^\circ \) from both sides:
\[ 3x = 360^\circ - 180^\circ \]
\[ 3x = 180^\circ \]
Now, divide both sides by 3:
\[ x = \frac{180^\circ}{3} \]
\[ x = 60^\circ \]

### Conclusion
The measure of the angle is \( 60^\circ \).