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Parallel and Perpendicular Lines

Parallel lines have the same slope and will never intersect. Parallel lines continue, literally, forever without touching (assuming that these lines are on the same plane).

Parallel Lines in greater depth

On the other hand, the slope of perpendicular lines are the negative reciprocals of each other, and a pair of these lines intersects at 90 degrees.

Perpendicular Lines in greater depth

Parallel Example

As you can see from the diagram below, these lines:

  • have the same slope
    • 2
  • are never going to intersect
Diagram of two parallel lines with equal slope

Perpendicular Example

As you can see from the picture below:

Parallel Lines in Greater Depth

Both of the lines below have the same slope: $$ \frac{1}{2} $$

Therefore these lines are parallel and will never meet.

Example of two parallel lines
Problem 1

Are the two lines pictured below parallel?

Classify these lines as parallel or not


While these lines may look parallel at first glance, if you look closely and calculate each line's slope, you will find that:

  • line 1 has a slope of $$ -\frac{1}{4} $$
  • line 2 's slope is $$ - \frac{1}{3} $$
Problem 2

Are the two lines below parallel?

Parall Lines Image

Yes these lines have the same slope, 2, and clearly are never going to intersect.

Perpendicular Lines in Greater Depth

Perpendicular lines:

  • intersect at $$ 90^{\circ} $$
  • have slopes that are negative reciprocals
Perpendicular lines
Problem 3

Are the two lines in the picture below perpendicular?


The slope of line 1 is -2 and that of line 2 is $$ \frac{1}{2}$$.

-2 and $$ \frac{1}{2}$$ are negative reciprocals so the lines are perpendicular.

Problem 4

Determine whether or not lines 6 and 5 perpendicular?

Diagram of Perpendicular Lines


Line 5's slope is $$ \frac{3}{2} $$ and line 6 's is $$ -\frac{2}{3} $$, which are negative reciprocals.

Problem 5

Are line P and line L perpendicular?

Example of Perpendicular Lines


They are not perpendicular because their slopes are not negative reciprocals of one another.

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