Parallel and Perpendicular Lines

Algebra 1 · Axiom Academy

LESSON Parallel & Perpendicular Lines The secret that decides whether two lines run side by side forever or cross at a perfect right angle is hiding in one number: the slope. 1. Parallel Lines: One Shared Slope Parallel lines never meet — they stay the same distance apart forever, like two train tracks. For that to happen they must climb at the exact same rate: equal slope . Below, L_2 starts on top of L_1 and slides straight down. Because we never tilt it, the slope stays and the gap between the lines never closes. Both lines rise 2 for every 3 they run Equal slopes → the lines stay parallel 2. Perpendicular Lines: The 90° Turn Perpendicular lines cross at a perfect right angle. To build one, take L_1 with slope m_1 = 2 and rotate it about the crossing point. Watch its rise/run triangle: the rise and run trade places and one of them flips sign, so a slope of turns into . The new slope is the negative reciprocal of the old one. Run 1, rise 2. The line climbs steeply to the right: . Turning the line a quarter-turn swaps the rise and the run and flips a sign. Run 2, rise -1 . Now the line falls gently: . Flip to (reciprocal) and negate it: . "Negative reciprocal" packs two moves into one: turn the slope upside down, then change its sign. Steep lines ( m_1 = 2 ) become gentle ones ( ), and lines going up become lines going down.

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