An angle is `60^(@)` more than one fifth of its complement. Find the smaller angle in degrees.
A `65^(@)`
B `35^(@)`
C `25^(@)`
D` 45^(@)`

Options

A. C
B. B
C. D
D. A

Answer

To solve the problem step by step, we can follow these instructions:

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

### Step 2: Define the complement
The complement of the angle \( x \) is given by \( 90 - x \) degrees.

### Step 3: Set up the equation
According to the problem, the angle \( x \) is 60 degrees more than one-fifth of its complement. This can be expressed mathematically as:
\[ x = \frac{1}{5}(90 - x) + 60 \]

### Step 4: Simplify the equation
First, distribute \( \frac{1}{5} \) on the right side:
\[ x = \frac{90}{5} - \frac{x}{5} + 60 \]
\[ x = 18 - \frac{x}{5} + 60 \]
Combine the constants:
\[ x = 78 - \frac{x}{5} \]

### Step 5: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 5:
\[ 5x = 5(78) - x \]
\[ 5x = 390 - x \]

### Step 6: Solve for \( x \)
Now, add \( x \) to both sides:
\[ 5x + x = 390 \]
\[ 6x = 390 \]
Now, divide both sides by 6:
\[ x = \frac{390}{6} = 65 \]

### Step 7: Find the complement
Now that we have \( x = 65 \) degrees, we can find its complement:
\[ 90 - x = 90 - 65 = 25 \]

### Step 8: Identify the smaller angle
The two angles we have are \( 65 \) degrees and \( 25 \) degrees. The smaller angle is:
\[ 25 \text{ degrees} \]

### Conclusion
Thus, the smaller angle is \( 25 \) degrees, which corresponds to option C.