Parallel and perpendicular slopes, with the vertical-line exception
Parallel nonvertical lines share a slope. Perpendicular nonvertical lines have slopes whose product is −1—but vertical and horizontal lines need separate language.
What the idea means, why its conditions matter, and where it connects.
Updated July 13, 2026The negative-reciprocal rule applies when both slopes are defined and nonzero.
Parallel means equal direction
Distinct parallel lines rise and run at the same rate, so their slopes match. Their intercepts differ, which keeps the lines from being identical.
Two vertical lines are also parallel even though neither has a numerical slope.
Perpendicular means a quarter turn
For ordinary nonvertical lines, rotating the direction by ninety degrees swaps rise and run and reverses one sign. That creates the negative reciprocal.
Do not merely change the sign or merely take the reciprocal; both changes matter.
Handle zero and undefined slopes directly
A horizontal line has slope zero, so its perpendicular partner is vertical and has undefined slope. The reciprocal formula cannot divide by zero.
Likewise, a vertical line is perpendicular to a horizontal line. State the equation x = constant or y = constant instead of inventing an infinite slope.
Find the line perpendicular to y = −3x + 7 through (6, 2).
Take the negative reciprocal.
Use point-slope form through the new point.
Simplify.
Verify perpendicular slopes.
Common mistakes
- Changing only the sign of the slope.
- Applying the reciprocal rule to slope zero.
- Using the original line's intercept for the new line.
Three takeaways
- Parallel lines share direction.
- Perpendicular slopes are negative reciprocals when defined.
- Horizontal and vertical lines are a special pair.