9.3 Angle Between Two Lines

Given $\theta = 45^\circ$ (so $\tan \theta = 1$) and $m_1 = 1/2$. Let $m_2 = m$. Formula: $\tan \theta = |\frac{m_2 - m_1}{1 + m_1 m_2}|$ $1 = \left| \frac{m - 1/2}{1 + m/2} \right|$ $1 = \left| \frac{2m - 1}{2 + m} \right|$ Two cases:

1. $\frac{2m - 1}{2 + m} = 1 \implies 2m - 1 = 2 + m \implies m = 3$.
2. $\frac{2m - 1}{2 + m} = -1 \implies 2m - 1 = -2 - m \implies 3m = -1 \implies m = -1/3$. Answer: Slope can be $3$ or $-1/3$.