parallel and perpendicular slope conditions
Two of the most-asked questions about a pair of lines are: do they run side by side forever (parallel), or do they cross at a perfect right angle (perpendicular)? Once you know each line's slope, both questions become a one-line check, with no protractor required.
Parallel lines never meet, so they must tilt the same way at the same steepness: two non-vertical lines are parallel exactly when their slopes are equal, m_1 = m_2 (and they are different lines, not the same one). Perpendicular lines cross at 90 degrees, and the rule is more surprising: two non-vertical lines are perpendicular exactly when their slopes are negative reciprocals, m_1 * m_2 = -1, equivalently m_2 = -1/m_1. So a line of slope 3 is perpendicular to a line of slope -1/3; a line of slope 2/5 is perpendicular to one of slope -5/2. Geometrically, turning a direction by 90 degrees swaps rise and run and flips one sign, which is exactly what taking the negative reciprocal does.
There is one boundary case the formulas cannot cover, and it is honest to state it: a vertical line has undefined slope, so the product rule cannot be applied. A vertical line (undefined slope) and a horizontal line (slope 0) are perpendicular, and two vertical lines are parallel — handle these by inspection rather than by m_1 * m_2 = -1. These two checks are the backbone of coordinate proofs about rectangles, right angles, altitudes, and perpendicular bisectors.
Lines y = 2x + 1 and y = 2x - 4 both have slope 2, so they are parallel. The line y = 2x + 1 and the line y = -(1/2)x + 7 have slopes 2 and -1/2, whose product is 2 * (-1/2) = -1, so they meet at a right angle — perpendicular.
Equal slopes mean parallel; slopes whose product is -1 mean perpendicular.
Perpendicular means negative reciprocal, not just negative: the perpendicular to slope 2 is -1/2, not -2. And the rule m_1 * m_2 = -1 silently assumes neither line is vertical — handle vertical/horizontal pairs separately.