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# Parallel and Skew Lines

## Lines that never intersect.

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Parallel and Skew Lines

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.

### Guidance

Two or more lines are parallel when they lie in the same plane and never intersect. The symbol for parallel is \begin{align*}||\end{align*} . To mark lines parallel, draw arrows \begin{align*}(>)\end{align*} on each parallel line. If there are more than one pair of parallel lines, use two arrows \begin{align*}(>>)\end{align*} for the second pair. The two lines below would be labeled \begin{align*}\overleftrightarrow{AB} \ || \ \overleftrightarrow{MN}\end{align*} or \begin{align*}l \ || \ m\end{align*} .

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

Skew lines are lines that are in different planes and never intersect. The difference between parallel lines and skew lines is parallel lines lie in the same plane while skew lines lie in different planes.

A transversal is a line that intersects two distinct lines. These two lines may or may not be parallel. The area between \begin{align*}l\end{align*} and \begin{align*}m\end{align*} is the called the interior . The area outside \begin{align*}l\end{align*} and \begin{align*}m\end{align*} is called the exterior .

The Parallel Lines Property is a transitive property that can be applied to parallel lines. It states that if lines \begin{align*}l \ || \ m\end{align*} and \begin{align*}m \ || \ n\end{align*} , then \begin{align*}l \ || \ n\end{align*} .

#### Example A

Are lines \begin{align*}q\end{align*} and \begin{align*}r\end{align*} parallel?

First find if \begin{align*}p \ || \ q\end{align*} , followed by \begin{align*}p \ || \ r\end{align*} . If so, then \begin{align*}q \ || \ r\end{align*} .

\begin{align*}p \ || \ q\end{align*} by the Converse of the Corresponding Angles Postulate, the corresponding angles are \begin{align*}65^\circ\end{align*} . \begin{align*}p \ || \ r\end{align*} by the Converse of the Alternate Exterior Angles Theorem, the alternate exterior angles are \begin{align*}115^\circ\end{align*} . Therefore, by the Parallel Lines Property, \begin{align*}q \ || \ r\end{align*} .

#### Example B

In the cube below, list 3 pairs of parallel planes.

Planes \begin{align*}ABC\end{align*} and \begin{align*}EFG\end{align*} , Planes \begin{align*}AEG\end{align*} and \begin{align*}FBH\end{align*} , Planes \begin{align*}AEB\end{align*} and \begin{align*}CDH\end{align*}

#### Example C

In the cube below, list 3 pairs of skew line segments.

\begin{align*}\overline{BD}\end{align*} and \begin{align*}\overline{CG}, \ \overline{BF}\end{align*} and \begin{align*}\overline{EG}, \ \overline{GH}\end{align*} and \begin{align*}\overline{AE}\end{align*} (there are others, too)

Watch this video for help with the Examples above.

### Vocabulary

Two or more lines are parallel when they lie in the same plane and never intersect. Skew lines are lines that are in different planes and never intersect. A transversal is a line that intersects two distinct lines.

### 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).

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

1. Which of the following is the best example of parallel lines?
2. Lamp Post and a Sidewalk
3. Longitude on a Globe
4. Stonehenge (the stone structure in Scotland)
2. Which of the following is the best example of skew lines?
1. Roof of a Home
2. Northbound Freeway and an Eastbound Overpass
3. Longitude on a Globe
4. The Golden Gate Bridge

For 3-10, determine whether the statement is true or false.

1. If \begin{align*}p || q\end{align*} and \begin{align*} q || r\end{align*} , then \begin{align*} p || r\end{align*} .
2. Skew lines are never in the same plane.
3. Skew lines can be perpendicular.
4. Planes can be parallel.
5. Parallel lines are never in the same plane.
6. Skew lines never intersect.
7. Skew lines can be in the same plane.
8. Parallel lines can intersect.
9. Come up with your own example of parallel lines in the real world.
10. Come up with your own example of skew lines in the real world.
11. What type of shapes do you know that have parallel line segments in them?
12. What type of objects do you know that have skew line segments in them?
13. If two lines segments are not in the same plane, are they skew?

### Vocabulary Language: English

Parallel

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

Skew

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

transversal

A transversal is a line that intersects two other lines.