# 2.5: Parallel, Perpendicular, and Skew Lines

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify parallel lines, skew lines and perpendicular lines.

## Parallel Lines and Planes

Parallel lines are coplanar (they lie in the same plane) and never intersect.

Below is an example of two parallel lines:

Parallel lines never ______________________________ each other.

Coplanar lines are in the same __________________________.

We use the symbol \begin{align*}\|\end{align*} for parallel, so we describe the figure above by writing \begin{align*}\overleftrightarrow{MN} \ \| \ \overleftrightarrow{CD}\end{align*}.

When we draw a pair of parallel lines, we use an arrow mark ( > ) on the lines to show that the lines are parallel. Just like with congruent segments, if there are two (or more) pairs of parallel lines, we use one arrow ( > ) for one pair and two (or more) arrows ( ≫ ) for the other pair.

There are two types of symbols to show that lines are parallel:

• In a geometric statement, the symbol _____________ is put in between two lines (“line \begin{align*}XY\end{align*}” is written as \begin{align*}\overleftrightarrow{XY}\end{align*}) to give the notation for parallel lines.
• In a picture, we draw the symbol _____________ on both lines to show that the lines are parallel to each other.

What symbols let you know that the lines below are parallel?

Fill in the blanks to make a symbolic statement that the two lines are parallel.

_____ \begin{align*}\|\end{align*} _____

## Perpendicular Lines

Perpendicular lines intersect at a \begin{align*}90^\circ\end{align*} right angle. This intersection is usually shown by a small square box in the \begin{align*}90^\circ\end{align*} angle.

Perpendicular lines meet at a ________________ angle.

The symbol \begin{align*}\bot\end{align*} is used to show that two lines, segments, or rays are perpendicular. In the picture above, we could write \begin{align*}\overrightarrow{BA} \ \bot \ \overleftrightarrow{BC}\end{align*}. (Notice that \begin{align*}\overrightarrow{BA}\end{align*} is a ray while \begin{align*}\overleftrightarrow{BC}\end{align*} is a line.)

Note that although "parallel" and "perpendicular" are defined in terms of lines, the same definitions apply to rays and segments with the minor adjustment that two segments or rays are parallel (or perpendicular) if the lines that contain the segments or rays are parallel (or perpendicular).

If you think about a table, the top of the table and the floor below it are usually in parallel planes.

## Skew Lines

The other of relationship you need to understand is skew lines. Skew lines are lines that are non-coplanar (they do not lie in the same plane) and never intersect.

Skew lines are in different _____________________ and never ______________________.

Segments and rays can also be skew. In the cube below, segment \begin{align*}\overline{AB}\end{align*} and segment \begin{align*}\overline{CG}\end{align*} are skew:

In the picture to the right...

• Put arrows on two line segments to show they are parallel
• Put a small square box at the intersection of two perpendicular segments
• Circle two line segments that are skew.

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