Find the acute angle between lines `x + y = 0` and `y = 0`

Answer

To find the acute angle between the lines \(x + y = 0\) and \(y = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The two lines given are:
1. \(x + y = 0\)
2. \(y = 0\)

### Step 2: Convert the equations to slope-intercept form
To find the slopes of the lines, we can rewrite them in the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

For the first line \(x + y = 0\):
\[ y = -x \]
Here, the slope \(m_1 = -1\).

For the second line \(y = 0\):
\[ y = 0 \cdot x + 0 \]
Here, the slope \(m_2 = 0\).

### Step 3: Use the formula for the angle between two lines
The formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

### Step 4: Substitute the slopes into the formula
Substituting \(m_1 = -1\) and \(m_2 = 0\) into the formula:
\[ \tan \theta = \left| \frac{0 - (-1)}{1 + (-1)(0)} \right| = \left| \frac{1}{1} \right| = 1 \]

### Step 5: Determine the angle
Now, we need to find the angle \(\theta\) such that:
\[ \tan \theta = 1 \]
The angle whose tangent is 1 is:
\[ \theta = \frac{\pi}{4} \text{ radians} \quad \text{(or 45 degrees)} \]

### Conclusion
Thus, the acute angle between the lines \(x + y = 0\) and \(y = 0\) is \(\frac{\pi}{4}\) radians.

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