# 3.2: Parallel and Skew Lines

- Page ID
- 2189

Lines that never intersect.

**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 |
---|---|

\(\overleftrightarrow{AB} \parallel \overleftrightarrow{MN}\) | Line \(AB\) is parallel to line \(MN\) |

\(l\parallel m\) | Line \(l\) is parallel to line \(m\). |

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, \(\overline{AB}\) and \(\overline{FH}\) are skew and \(\overline{AC}\) and \(\overline{EF}\) are skew.

#### Basic Facts About Parallel Lines

__Property__: If lines \(l\parallel m\) and \(m\parallel n\), then \(l\parallel n\).

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 \(A\), but only

**that is**

*one*__parallel__to \(l\).

A **transversal** is a line that intersects two other lines. The area * between* \(l\) and \(m\) is the

*. The area*

*interior**\(l\) and \(m\) is the*

*outside**.*

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

Use the figure below for Examples \(\PageIndex{1}\) and \(\PageIndex{2}\). The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them.

Example \(\PageIndex{1}\)

Find two pairs of skew lines.

**Solution**

\(\overline{ZV}\) and \(\overline{WB}\). \(\overline{YD}\) and \(\overline{VW}\)

Example \(\PageIndex{2}\)

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

**Solution**

One line, \(\overline{CD}\)

Example \(\PageIndex{3}\)

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

**Solution**

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 \(\PageIndex{4}\)

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

**Solution**

One possible answer is lines \(\overline{AB}\) and \(\overline{EF}\).

Example \(\PageIndex{5}\)

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

**Solution**

One possible answer is \(\overline{BD}\) and \(\overline{CG}\).

### 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 \(m\angle 2=55^{\circ}\), what other angles do you know?
- If \(m\angle 5=123^{\circ}\), what other angles do you know?
- Is \(l\parallel m\)? Why or why not?

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

- If \(p\parallel q \)and \(q\parallel r\), then \(p\parallel r\).
- 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

## Vocabulary

Term | Definition |
---|---|

parallel lines |
Two or more lines that lie in the same plane and never intersect. Parallel lines will always have the same .slope |

Skew lines |
Skew lines are lines that are in different planes and never intersect. |

transversal |
A transversal is a line that intersects two other lines. |

Parallel |
Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope. |

Skew |
To skew a given set means to cause the trend of data to favor one end or the other |

## Additional Resources

Interactive Element

Video: Proving Lines Parallel

Practice: Parallel and Skew Lines Discussion Questions

Study Aids: Lines and Angles Study Guide

Pratice: Parallel and Skew Lines

Real World: Short Circuits: How Parallel Circuits Work