What if you were given a pair of lines that never intersect and were asked to describe them? What terminology would you use? After completing this Concept, you will be able to define the terms parallel line, skew line, and transversal. You'll also be able to apply the properties associated with parallel lines.

### Watch This

Watch the portions of this video dealing with parallel lines.

Then watch this video.

James Sousa: Parallel Line Postulate

### Guidance

**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*

##### Transversals

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*.

#### Example A

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 B

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 C

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*}.

### Guided Practice

Use the figure below to answer the questions. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them.

1. Find two pairs of skew lines.

2. List a pair of parallel lines.

3. 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).

**Answers:**

1. \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*}

2. \begin{align*}\overline{ZV}\end{align*} and \begin{align*}\overline{EA}\end{align*}.

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

### Practice

- 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.