The measure of an angle whose supplement is three times as large as its complement, is

Options

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

Answer

To solve the problem of finding the measure of an angle whose supplement is three times as large as its complement, we can follow these steps:

### Step 1: Define the angle
Let the angle be denoted as \( \theta \).

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

### Step 3: Set up the equation based on the problem statement
According to the problem, the supplement is three times the complement:
\[ 180^\circ - \theta = 3(90^\circ - \theta) \]

### Step 4: Expand and simplify the equation
Expanding the right side:
\[ 180^\circ - \theta = 270^\circ - 3\theta \]

### Step 5: Rearrange the equation
Now, we will rearrange the equation to isolate \( \theta \):
\[ 180^\circ - \theta + 3\theta = 270^\circ \]
This simplifies to:
\[ 180^\circ + 2\theta = 270^\circ \]

### Step 6: Solve for \( \theta \)
Subtract \( 180^\circ \) from both sides:
\[ 2\theta = 270^\circ - 180^\circ \]
\[ 2\theta = 90^\circ \]
Now, divide both sides by 2:
\[ \theta = \frac{90^\circ}{2} = 45^\circ \]

### Conclusion
Thus, the measure of the angle is:
\[ \theta = 45^\circ \]