An angle is `30^(@)` more than one half of its complement . Find difference between the greater and the samaller angles
A. `10^(@)`
B. `20^(@)`
C. ` 30^(@)`
D. `25^(@)`

Options

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

Answer

To solve the problem, we need to find the angle that is 30 degrees more than one half of its complement. Let's break this down step by step.

### Step 1: Define the angles
Let the angle be \( x \). The complement of \( x \) is \( 90 - x \).

### Step 2: Set up the equation
According to the problem, the angle \( x \) is 30 degrees more than one half of its complement. We can express this mathematically as:
\[ x = \frac{1}{2}(90 - x) + 30 \]

### Step 3: Simplify the equation
Now, let's simplify the equation:
\[ x = \frac{90 - x}{2} + 30 \]
Multiply the entire equation by 2 to eliminate the fraction:
\[ 2x = 90 - x + 60 \]
Combine like terms:
\[ 2x + x = 150 \]
\[ 3x = 150 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 3:
\[ x = \frac{150}{3} = 50 \]

### Step 5: Find the complement
Now that we have \( x \), we can find the complement:
\[ y = 90 - x = 90 - 50 = 40 \]

### Step 6: Find the difference between the angles
Now we need to find the difference between the greater angle \( x \) and the smaller angle \( y \):
\[ \text{Difference} = x - y = 50 - 40 = 10 \]

### Conclusion
Thus, the difference between the greater and the smaller angles is \( 10^{\circ} \).

### Final Answer
The correct option is A. \( 10^{\circ} \).

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