What are the conditions for parallel and perpendicular lines in terms of their slopes?

If two lines are parallel, the angle $\theta$ between them is $0^\circ$. This means that their slopes are equal.

$$m_1 = m_2$$

If two lines are perpendicular, the angle $\theta$ between them is $90^\circ$. The relationship between their slopes $m_1$ and $m_2$ is derived from the formula for the angle between two lines, $\tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2}$. Since $\tan 90^\circ$ is undefined, the denominator must be zero.

$$1 + m_1 m_2 = 0$$

To find the relationship between $m_1$ and $m_2$, we rearrange the equation:

$$m_1 m_2 = -1$$

This means that the product of the slopes of two perpendicular lines is $-1$. Alternatively, one slope is the negative reciprocal of the other:

$$m_1 = \frac{-1}{m_2}$$