Two complementary angles differ by `8^(@)` . Find the angles.

Options

A. `40^(@),50^(@)`
B. `40^(@),60^(@)`
C. `41^(@),49^(@)`
D. `40^(@),48^(@)`

Answer

To solve the problem of finding two complementary angles that differ by 8 degrees, we can follow these steps:

### Step 1: Define the Variables
Let the first angle be \( x \) degrees and the second angle be \( y \) degrees.

### Step 2: Set Up the Equations
Since the angles are complementary, their sum is:
\[ x + y = 90 \quad \text{(1)} \]
We are also given that the difference between the two angles is:
\[ x - y = 8 \quad \text{(2)} \]

### Step 3: Solve the Equations
Now we have a system of two equations. We can solve these equations simultaneously.

**Add equations (1) and (2):**
\[ (x + y) + (x - y) = 90 + 8 \]
This simplifies to:
\[ 2x = 98 \]
Now, divide both sides by 2:
\[ x = 49 \]

### Step 4: Find the Value of \( y \)
Now that we have \( x \), we can substitute it back into equation (1) to find \( y \):
\[ 49 + y = 90 \]
Subtract 49 from both sides:
\[ y = 90 - 49 \]
Thus,
\[ y = 41 \]

### Step 5: Conclusion
The two complementary angles are:
\[ x = 49^\circ \quad \text{and} \quad y = 41^\circ \]

### Summary of the Solution
The angles are \( 49^\circ \) and \( 41^\circ \).

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