### Parallel and Skew Lines

**Parallel** lines are two or more lines that lie in the same plane and never intersect. To show that lines are parallel, arrows are used.

Label It |
Say It |
---|---|

\begin{align*}\overleftrightarrow{AB} || \overleftrightarrow{MN}\end{align*} | Line \begin{align*}AB\end{align*} is parallel to line \begin{align*}MN\end{align*} |

\begin{align*}l || m\end{align*} | Line \begin{align*}l\end{align*} is parallel to line \begin{align*}m\end{align*}. |

In the definition of parallel the word “line” is used. However, line segments, rays and planes can also be parallel. The image below shows two parallel planes, with a third blue plane that is perpendicular to both of them.

**Skew** lines are lines that are in different planes and never intersect. They are different from parallel lines because parallel lines lie in the SAME plane. In the cube below, \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{FH}\end{align*} are skew and \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{EF}\end{align*} are skew.

#### Basic Facts About Parallel Lines

Property: If lines \begin{align*}l || m\end{align*} and \begin{align*}m || n\end{align*}, then \begin{align*}l || n\end{align*}.

If then

Postulate: For any line and a point ** not** on the line, there is one line parallel to this line through the point. There are infinitely many lines that go through \begin{align*}A\end{align*}, but only

**that is parallel to \begin{align*}l\end{align*}.**

*one*

A **transversal** is a line that intersects two other lines. The area *between* \begin{align*}l\end{align*} and \begin{align*}m\end{align*} is the *interior*. The area *outside* \begin{align*}l\end{align*} and \begin{align*}m\end{align*} is the *exterior*.

What if you were given a pair of lines that never intersect and were asked to describe them? What terminology would you use?

### Examples

Use the figure below for Examples 1 and 2. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them.

#### Example 1

Find two pairs of skew lines.

\begin{align*}\overline{ZV}\end{align*} and \begin{align*}\overline{WB}\end{align*}. \begin{align*}\overline{YD}\end{align*} and \begin{align*}\overline{VW}\end{align*}

#### Example 2

For \begin{align*}\overline{XY}\end{align*}, how many parallel lines would pass through point \begin{align*}D\end{align*}? Name this/these line(s).

One line, \begin{align*}\overline{CD}\end{align*}

#### Example 3

True or false: some pairs of skew lines are also parallel.

This is false, by definition skew lines are in **different** planes and parallel lines are in the **same** plane. Two lines could be skew or parallel (or neither), but never both.

#### Example 4

Using the cube below, list a pair of parallel lines.

One possible answer is lines \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{EF}\end{align*}.

#### Example 5

Using the cube below, list a pair of skew lines.

One possible answer is \begin{align*}\overline{BD}\end{align*} and \begin{align*}\overline{CG}\end{align*}.

### Review

- Which of the following is the best example of parallel lines?
- Railroad Tracks
- Lamp Post and a Sidewalk
- Longitude on a Globe
- Stonehenge (the stone structure in Scotland)

- Which of the following is the best example of skew lines?
- Roof of a Home
- Northbound Freeway and an Eastbound Overpass
- Longitude on a Globe
- The Golden Gate Bridge

Use the picture below for questions 3-5.

- If \begin{align*}m\angle 2 = 55^\circ\end{align*}, what other angles do you know?
- If \begin{align*}m\angle 5 = 123^\circ\end{align*}, what other angles do you know?
- Is \begin{align*}l || m\end{align*}? Why or why not?

For 6-10, determine whether the statement is true or false.

- If \begin{align*}p || q\end{align*} and \begin{align*} q || r\end{align*}, then \begin{align*} p || r\end{align*}.
- Skew lines are never in the same plane.
- Skew lines can be perpendicular.
- Planes can be parallel.
- Parallel lines are never in the same plane.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.1.

### Resources