Parallel and Perpendicular Lines

Two lines are parallel if they have the same slope but different y-intercepts.

$$\text{Parallel: } m_1 = m_2 \text{ and } b_1 \neq b_2$$

Parallel lines never intersect.

Example: $y = 3x + 1$ and $y = 3x - 5$ are parallel (both have slope $3$).

Two lines are perpendicular if their slopes are negative reciprocals of each other:

$$m_1 \times m_2 = -1 \quad \text{or equivalently} \quad m_2 = -\frac{1}{m_1}$$

Perpendicular lines intersect at a 90$^{\circ}$ angle.

Examples:
- Slopes $2$ and $-\frac{1}{2}$: $2 \times (-\frac{1}{2}) = -1$ ✓ perpendicular
- Slopes $\frac{3}{4}$ and $-\frac{4}{3}$: $\frac{3}{4} \times (-\frac{4}{3}) = -1$ ✓ perpendicular
- Slopes $3$ and $\frac{1}{3}$: $3 \times \frac{1}{3} = 1 \neq -1$ ✗ NOT perpendicular