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Understanding Slopes of Parallel and Perpendicular Lines

Explore the relationships between the slopes of parallel and perpendicular lines in this informative guide. Discover that parallel lines share the same slope, while perpendicular lines have slopes that are opposites and reciprocals (assuming neither line is vertical). Through detailed examples, you'll learn to determine the slopes of lines given their relationships and understand special cases involving vertical and horizontal lines. This summary emphasizes the key concepts essential for mastering the geometry of lines in a plane.

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Understanding Slopes of Parallel and Perpendicular Lines

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  1. Slopes of Parallel and Perpendicular Lines Recall that two lines are parallel if they are in the same plane and do not intersect. Two lines are perpendicular if they intersect at a right angle

  2. Example 1: Consider the following graph with two lines that are parallel. Since both lines have the same “steepness”, their slopes must be the same value.

  3. Example 2: Consider the following graph with two lines that are perpendicular. Notice that the line in red has a positive slope (rising), while the line in blue has a negative slope (falling).

  4. This is a quite interesting result. Not only are the slopes opposite in sign, but they are reciprocals! As it turns out, this is always the case, assuming that neither line is vertical, which has undefined slope.

  5. Example 3: Special case #1 When both lines are vertical, they are parallel. Recall that vertical lines have undefined slope. Thus, …

  6. Example 4: Special case #2 When one line is vertical, and the other is horizontal, then the lines are perpendicular. The slope of the vertical line is undefined … … and the slope of the horizontal line is zero.

  7. SUMMARY (assume no vertical lines) Parallel lines Perpendicular lines Slopes are opposite reciprocals Slopes are the same

  8. Example 5: Line 1 has a slope of – 3. a) Determine the slope of Line 2 that is parallel to Line 1 b) Determine the slope of Line 3 if it is perpendicular to Line 1

  9. Example 6: Line 1 has a slope of 3/4. a) Determine the slope of Line 2 that is parallel to Line 1

  10. Line 1 has a slope of 3/4. b) Determine the slope of Line 3 if it is perpendicular to Line 1

  11. While we have written out all the details, these problems are usually done in your head. For part b again … Use two steps: 1) The reciprocal is: 2) Take the opposite:

  12. END OF PRESENTATION

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