To find the angle between two lines, we use the formula:

$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$

where $m_1$ and $m_2$ are the slopes of the two lines.

Step 1: Find the slopes of the given lines.

The equation of the first line is $x + y = 0$. Rearranging into slope-intercept form ($y = mx + b$):

$y = -x \quad \Rightarrow \quad m_1 = -1$

The equation of the second line is $x - y = 0$. Rearranging it similarly:

$y = x \quad \Rightarrow \quad m_2 = 1$

Step 2: Use the formula for the angle.

Now, substitute $m_1 = -1$ and $m_2 = 1$ into the angle formula:

$\tan \theta = \left| \frac{-1 - 1}{1 + (-1)(1)} \right|$

Simplifying:

$\tan \theta = \left| \frac{-2}{1 - 1} \right| = \left| \frac{-2}{0} \right|$

Since division by zero is undefined, this means the two lines are perpendicular.

Conclusion:

The angle between the two lines is $90^\circ$.

Let me know if you'd like any further clarification or details!

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Here are some follow-up questions to explore:

1. How do you find the angle between two non-perpendicular lines?
2. What does it mean geometrically for two lines to be perpendicular?
3. How would you calculate the angle between two lines in three-dimensional space?
4. How can you determine if two lines are parallel from their equations?
5. What are the steps to finding the angle between two curves?

Tip: When the denominator of the tangent formula is zero, the lines are perpendicular.