A line makes `45^@` with positive x-axis and makes equal angles with positive y,z axes, respectively. What is the sum of the three angles which the line makes with positive x, y and z axes ?

Options

A. `180^@`
B. `165^@`
C. `150^@`
D. `135^@`

Explanation

To solve the problem step by step, we need to find the sum of the angles that a line makes with the positive x, y, and z axes, given that it makes an angle of \(45^\circ\) with the positive x-axis and equal angles with the positive y and z axes.

### Step 1: Define the angles
Let:
- The angle with the x-axis be \( \alpha = 45^\circ \)
- The angle with the y-axis be \( \beta \)
- The angle with the z-axis be \( \gamma \)

Since the line makes equal angles with the y and z axes, we can say:
\[ \beta = \gamma \]

### Step 2: Use the property of direction cosines
The direction cosines of the angles made by a line with the coordinate axes satisfy the equation:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Step 3: Substitute the known values
Substituting \( \alpha = 45^\circ \) into the equation:
\[ \cos^2(45^\circ) + \cos^2(\beta) + \cos^2(\gamma) = 1 \]
Since \( \beta = \gamma \), we can rewrite it as:
\[ \cos^2(45^\circ) + 2\cos^2(\beta) = 1 \]

### Step 4: Calculate \( \cos^2(45^\circ) \)
We know that:
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 5: Substitute and solve for \( \cos^2(\beta) \)
Now substitute \( \cos^2(45^\circ) \) into the equation:
\[ \frac{1}{2} + 2\cos^2(\beta) = 1 \]
Subtract \( \frac{1}{2} \) from both sides:
\[ 2\cos^2(\beta) = 1 - \frac{1}{2} = \frac{1}{2} \]
Dividing both sides by 2:
\[ \cos^2(\beta) = \frac{1}{4} \]

### Step 6: Find \( \cos(\beta) \)
Taking the square root:
\[ \cos(\beta) = \frac{1}{2} \]
Thus, the angle \( \beta \) is:
\[ \beta = 60^\circ \]

### Step 7: Find \( \gamma \)
Since \( \beta = \gamma \):
\[ \gamma = 60^\circ \]

### Step 8: Calculate the sum of the angles
Now, we can find the sum of the angles:
\[ \alpha + \beta + \gamma = 45^\circ + 60^\circ + 60^\circ = 165^\circ \]

### Final Answer
The sum of the three angles which the line makes with the positive x, y, and z axes is:
\[ \boxed{165^\circ} \]