Parallel and Perpendicular Lines

Two Special Relationships

When two lines share the plane, two arrangements stand out:

• Parallel: they run side by side and never meet
• Perpendicular: they cross at a perfect right angle

Slope alone tells you which one you have.

Parallel: Same Slope

Two lines are parallel exactly when they have the same slope. Same steepness, same direction, just shifted apart.

Parallel lines: $m_1 = m_2$

For example, $y = 2x + 1$ and $y = 2x - 4$ are parallel. Both climb at slope $2$; they simply cross the $y$-axis at different heights.

Perpendicular: Negative Reciprocals

Right angles are less obvious. Two lines are perpendicular when their slopes are negative reciprocals of each other.

As one line turns, the other turns with it, always locked at $90°$. Look at the readout: the two slopes are never equal, but their product is always $-1$.

Perpendicular lines: $m_1 \cdot m_2 = -1$, so $m_2 = -\dfrac{1}{m_1}$

To find a perpendicular slope, do two things: flip the fraction and change the sign.

The One Exception

A horizontal line (slope $0$) is perpendicular to a vertical line (undefined slope).

The product rule breaks down here, because you cannot multiply by an undefined slope. But geometrically it is obvious: flat and upright meet at a right angle.

Example

Is the line through $(0, 1)$ and $(2, 5)$ perpendicular to a line of slope $-\frac{1}{2}$?

First find the slope of the first line:

$m_1 = \frac{5 - 1}{2 - 0} = 2$

Now check the product:

$m_1 \cdot m_2 = 2 \cdot \left(-\tfrac{1}{2}\right) = -1$

The product is $-1$, so yes, the lines are perpendicular.