Two lines passing through (3, 2) make an angle of 45°. If the slope of one of the lines is 1, find the slope of the other line.

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?

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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.

Discover how to find the slope of a line passing through (3, 2) when the angle between the lines is 45°. This problem involves understanding slopes, the angle between lines formula, and solving equations. Suitable for high school students in Grades 10-12.

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