Parallel and Perpendicular Lines

Geometry · Axiom Academy

LESSON Parallel and Perpendicular Lines How the slopes of two lines reveal whether they run parallel, cross at a right angle, or just meet. 1. Parallel Lines Have Equal Slopes Two non-vertical lines are parallel if and only if they have the same slope . They rise at exactly the same rate, so they stay the same distance apart forever and never intersect. Equal slopes the lines are parallel 2. Perpendicular Lines: Slopes Multiply to −1 Two non-vertical lines are perpendicular if and only if the product of their slopes is −1 . Equivalently, the slopes are negative reciprocals of each other, and the lines meet at a angle. Product of slopes is −1 the lines are perpendicular Each slope is the negative reciprocal of the other 3. Finding the Perpendicular Slope To get the slope of a perpendicular line, flip the fraction and change the sign — swap rise and run, then negate. The animation shows a slope triangle being tipped onto its side: a rise of 2 over a run of 3 becomes a rise of -3 over a run of 2 . Flip to , negate → perpendicular slope . Write as , flip and negate → perpendicular slope . Multiply the two slopes; a correct negative reciprocal always gives -1 . 4. Example 1 — Are These Lines Parallel? Line 1 passes through (1, 3) and (4, 9) . Line 2 passes through (-2, 1) and (1, 7) . Are they parallel? Compute and compare the slopes 5. Example 2 — Are These Lines Perpendicular? Line 1 has equation y = 3x - 5 . Line 2 passes through (6, 2) and (9, 1) . Are they perpendicular?

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