Let the slopes of the two lines be $m_1$ and $m_2$, where $m_1 = 1$ is given for one of the lines, and we need to find the slope $m_2$ of the other line.

The formula to find the angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is:

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

Given that $\theta = 45^\circ$, we know that:

$\tan(45^\circ) = 1$

Thus, the equation becomes:

$1 = \left| \frac{m_2 - 1}{1 + 1 \cdot m_2} \right|$

We now have two cases to consider:

Case 1: Positive value

$\frac{m_2 - 1}{1 + m_2} = 1$ Solving for $m_2$: $m_2 - 1 = 1 + m_2$ $m_2 - m_2 = 2$ This gives a contradiction, so there is no solution in this case.

Case 2: Negative value

$\frac{m_2 - 1}{1 + m_2} = -1$ Solving for $m_2$: $m_2 - 1 = - (1 + m_2)$ $m_2 - 1 = -1 - m_2$ $2m_2 = 0$ $m_2 = 0$

Thus, the slope of the other line is $m_2 = 0$.

Final Answer:

The slope of the other line is 0.

Would you like further explanation on any step or more details?

---

Here are 5 related questions:

1. How is the formula for the angle between two lines derived?
2. What happens if the angle between the lines is $90^\circ$?
3. How would the equation change if the angle was different, say $60^\circ$?
4. Can two lines have the same slope and still make an angle between them?
5. What is the geometric interpretation of a line with a slope of zero?

Tip: When dealing with angles between lines, remember that the tangent of the angle gives the relationship between their slopes.