Find the measure of an angle, if six times its complement is `12^(@)` less than twice of its supplement

Options

A. `192^(@)`
B. `52^(@)`
C. `48^(@)`
D. none of these

Answer

To find the measure of the angle, let's follow these steps:

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

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

### Step 3: Set up the equation based on the problem statement
According to the problem, six times the complement is 12 degrees less than twice the supplement. This can be expressed as:
\[ 6 \times (90^\circ - x) = 2 \times (180^\circ - x) - 12^\circ \]

### Step 4: Expand both sides of the equation
Expanding the left side:
\[ 6 \times (90 - x) = 540 - 6x \]
Expanding the right side:
\[ 2 \times (180 - x) - 12 = 360 - 2x - 12 = 348 - 2x \]

### Step 5: Set the two sides of the equation equal to each other
Now we have:
\[ 540 - 6x = 348 - 2x \]

### Step 6: Rearrange the equation to isolate \( x \)
To isolate \( x \), first, move all terms involving \( x \) to one side and constant terms to the other side:
\[ 540 - 348 = -2x + 6x \]
This simplifies to:
\[ 192 = 4x \]

### Step 7: Solve for \( x \)
Now, divide both sides by 4:
\[ x = \frac{192}{4} = 48 \]

### Conclusion
The measure of the angle is:
\[ \boxed{48^\circ} \]

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