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Parallel and Skew Lines

Lines that never intersect.

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Parallel and Skew Lines

Parallel and Skew Lines 

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*}AB || MN or \begin{align*}l \ || \ m\end{align*}l || m.

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*}A, but only one is parallel to \begin{align*}l\end{align*}l.

Watch the portions of this video dealing with parallel lines.

 

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*}l and \begin{align*}m\end{align*}m is the called the interior. The area outside \begin{align*}l\end{align*}l and \begin{align*}m\end{align*}m 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*}l || m and \begin{align*}m \ || \ n\end{align*}m || n, then \begin{align*}l \ || \ n\end{align*}l || n.

 

Determining if Two Lines are Parallel 

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

Notice that the arrow markings indicate that \begin{align*}p \ || \ q\end{align*}p || q. Similarly, arrow markings indicate that \begin{align*}p \ || \ r\end{align*}p || r. This means that \begin{align*}q \ || \ r\end{align*}q || r by the Parallel Lines Property.

Identifying Parallel Lines 

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

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

Identifying Skew Line Segments 

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

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

 

 

 

Examples 

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.

Example 1

Find two pairs of skew lines.

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

Example 2

List a pair of parallel lines.

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

Example 3

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

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

Interactive Practice

 

 

 

Review

  1. Which of the following is the best example of parallel lines?
    1. Railroad Tracks
    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*}p||q and \begin{align*} q || r\end{align*}q||r, then \begin{align*} p || r\end{align*}p||r.
  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?

Review (Answers)

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

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Vocabulary

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

transversal

A transversal is a line that intersects two other lines.

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