Perpendicular and Parallel Lines

Perpendicular Slopes Multiply to Minus One

Two lines are perpendicular exactly when their slopes multiply to minus one. That is why the perpendicular slope is the negative reciprocal rather than simply the opposite, and why a slope of 2 has a perpendicular of minus one half rather than minus 2.

Working Through the Reason

If one line rises at 45 degrees, its perpendicular is also at 45 degrees but pointing the other way. Any slope can be read as a tangent of an angle, and since the two angles of a right angle sum to 90 degrees, their tangents must multiply to minus one.

Parallel Lines Have Equal Slopes

Two non vertical lines are parallel exactly when their slopes are equal. There is no sign change and no reciprocal involved, which makes parallel detection the simpler of the two tests.

The Vertical and Horizontal Exception

A vertical line has undefined slope and its perpendicular is horizontal with a slope of zero. The reciprocal rules break down here, so these two cases are handled separately rather than through the general formula.

The Same Intercept Means They Intersect

Two non parallel lines intersect exactly once. If they have the same slope and the same intercept they are the same line, and if they have the same slope but different intercepts they are parallel and never meet.

Checking Your Work

Multiply the two slopes. A product of minus one means perpendicular, a product equal to one positive means parallel, and anything else means the lines intersect at an angle you can find from the difference in their angles of inclination.