Parallel, Perpendicular, or Neither?

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Are two lines parallel, perpendicular, or neither? Use this article to learn how to use the slope of a linear function to answer this question.

Parallel Lines

Aerial view of linear road and blue sea.
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Characteristics of Parallel Lines

  • A set of parallel lines have the same slope.
  • A set of parallel lines never intersect.
  • Notation: Line A ll Line B (Line A is parallel to Line B.)

Note: Parallel lines are not automatically congruent; don't confuse length with slope.

Examples of Parallel Lines

  • The path of two cars driving eastbound on Interstate 10
  • Parallelograms: A parallelogram is comprised of four sides. Each side is parallel to its opposite side. Rectangles, squares, and rhombi (more than 1 rhombus) are parallelograms
  • Lines with the same slope (per the slope formula) — Line 1: m = -3; Line 2: m = -3
  • Lines with the same rise and run. Look at the picture above. Notice that the slope for each of these lines is -3/2
  • Lines with the same m, slope, in the equation. Example: y = 2x + 5; y = 10 + 2x

Note: Yes, parallel lines share a slope, but they cannot share a y-intercept. What would happen if the y-intercepts were the same?

Perpendicular Lines

Norwegian flag
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Characteristics of Perpendicular Lines

  • Perpendicular lines cross to form 90° angles at the intersection.
  • The slopes of perpendicular lines are negative reciprocals. To illustrate, the slope of Line F is 2/5. What is the slope of a line perpendicular to Line F? Flip over the slope and change the sign. The slope of the perpendicular line is -5/2.
  • The product of the slopes of perpendicular lines is -1. For example, 2/5 * -5/2 = -1.

Note: Each set of intersecting lines is not a set of perpendicular lines. Right angles must be formed at the intersection.

Examples of Perpendicular Lines

  • The blue stripes on the flag of Norway
  • The intersecting sides of rectangles and squares
  • The legs of a right triangle
  • Equations: y = -3x + 5; y = 1/3x + 5;
  • The result of the slope formula: m = 1/2; m = -2
  • Lines with slopes that are negative reciprocals. Look at the two lines in the picture. Notice that the slope of the upward sloping line is 5, yet the slope of the downward sloping line is -1/5


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Characteristics of Lines that are Neither Parallel nor Perpendicular

  • Slopes are not the same
  • The lines intersect
  • Although the lines intersect, they do not form 90° angles.

Examples of "Neither" Lines

  • The hour and minute hands of a clock at 10:10 p.m
  • The red stripes on the American Samoa flag