## Parallel and Perpendicular Lines

Read this page to see the special relationship of the slopes of parallel and perpendicular lines. After you read, try a few practice problems.

If you know the slopes of two non-vertical lines, can you easily determine if they are parallel? Perpendicular? Yes!

**Parallel lines** have the same slope.

**Perpendicular lines** have slopes that are opposite reciprocals.

Slopes of Parallel Lines

Two lines in a plane are **parallel** if they never intersect; they have the same **slant**. We can use slope (or lack of it) to decide if lines are parallel:

#### Characterization of Parallel Lines Involving Slope

Suppose two distinct lines lie in the same coordinate plane.

Then, the two lines are **parallel*** *if and only if **they are both vertical** or **they have the same slope**.

This statement offers a good opportunity to review **the mathematical words** **if and only if** and **or**.

- The basic set-up is two lines lying in the same coordinate plane. This condition lurks in the background (is assumed to be true) for all that follows.
- Two mathematical sentences are being compared with the words
**if and only if**:- The two lines are parallel,
- They are both vertical OR they have the same slope.

**If and only if**is a variation of**is equivalent to**. Sentences that are equivalent always have the same truth values – they are true at the same time, and false at the same time. If one is true, so is the other. If one is false, so is the other.- An
**or**sentence is true when at least one of the subsentences is true. The only time an**or**is false is when both subsentences are false. - The nice, compact statement above is actually saying A LOT.

Slopes of Perpendicular Lines

Now, let's move on to perpendicular lines. Two lines are **perpendicular** if they intersect at a 90^{ο} angle. For example, the x-axis and y-axis are perpendicular.

#### Characterization of Perpendicular Lines Involving Slope

Suppose two non-vertical lines lie in the same coordinate plane. Let and denote the slopes of these lines.

The two lines are **perpendicular** if and only if their slopes are opposite reciprocals:

Equivalently, the two lines are perpendicular if and only if their slopes multiply to

Make sure you understand the opposite reciprocal equation:

one of the slopes |
is |
the opposite of |
the reciprocal of the other slope |

For example, what is the reciprocal of ?
**Answer**:

What is the opposite [of the] reciprocal of ?
**Answer**:

- So, the numbers and are opposite reciprocals.
- Also notice that .
- So, lines with slopes and are perpendicular.

It is easy to see that this is the correct characterization for perpendicular lines, by studying the sketch below:

The yellow triangle, with base of length and right side of length , shows that the slope of the first line is .

Now, imagine that this yellow triangle is a block of wood that is glued to the line. Rotate this block of wood counter-clockwise by 90^{ο} (so the original base is now vertical). Using the rotated triangle to compute the slope of the new,
rotated, line gives:

Easy! Voila!

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/par_perpen.htm

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