The difference of two complementary angle is `40^@`. The angles are

Options

A. `65^@, 35^@`
B. `70^@, 30^@`
C. `25^@, 65^@`
D. `70^@, 110^@`

Answer

To solve the problem of finding two complementary angles where the difference between them is \(40^\circ\), we can follow these steps:

### Step 1: Define the angles
Let the two complementary angles be \(x\) and \(y\).

### Step 2: Set up the equations
Since the angles are complementary, we know that:
\[ x + y = 90^\circ \quad \text{(1)} \]
We are also given that the difference between the two angles is:
\[ x - y = 40^\circ \quad \text{(2)} \]

### Step 3: Solve the equations
Now we can solve these two equations simultaneously.

From equation (1), we can express \(y\) in terms of \(x\):
\[ y = 90^\circ - x \quad \text{(3)} \]

Now, substitute equation (3) into equation (2):
\[ x - (90^\circ - x) = 40^\circ \]
This simplifies to:
\[ x - 90^\circ + x = 40^\circ \]
\[ 2x - 90^\circ = 40^\circ \]
Adding \(90^\circ\) to both sides gives:
\[ 2x = 130^\circ \]
Dividing both sides by 2:
\[ x = 65^\circ \quad \text{(4)} \]

### Step 4: Find \(y\)
Now, substitute the value of \(x\) back into equation (3) to find \(y\):
\[ y = 90^\circ - 65^\circ = 25^\circ \quad \text{(5)} \]

### Step 5: Conclusion
Thus, the two complementary angles are:
\[ x = 65^\circ \quad \text{and} \quad y = 25^\circ \]

### Final Answer
The angles are \(65^\circ\) and \(25^\circ\).

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