Parallel & Perpendicular Lines

Parallel lines never meet, which means they have exactly the same slope. Different intercepts, or they would be the same line.

Perpendicular lines meet at 90°. Their slopes are negative reciprocals: flip the fraction and change the sign.

Slope 2/3 → perpendicular slope −3/2.
Slope −4 → perpendicular slope 1/4.

A quick check: multiply the two slopes. If you get −1, they are perpendicular.

Exception: a horizontal line (slope 0) and a vertical line (undefined slope) are perpendicular, even though that product does not exist.

Parallel lines have equal slopes because slope determines direction; equal slopes with equal intercepts give the same line rather than a parallel pair.

For perpendicularity, rotating a direction vector ⟨1, m⟩ by 90° gives ⟨−m, 1⟩, whose slope is −1/m. Hence m₁m₂ = −1 for non-vertical, non-horizontal lines. The horizontal/vertical pair is perpendicular but excluded from the product test, since an undefined slope cannot be multiplied.

Typical tasks combine this with point-slope form: given a line and a point, find the parallel or perpendicular line through that point. The slope comes from the relationship, the point from the problem, and point-slope assembles them.

Note that the coefficients in standard form make the relationship visible without conversion: Ax + By = C and Bx − Ay = D are always perpendicular, since their slopes −A/B and B/A multiply to −1.