# 2.5: Parallel, Perpendicular, and Skew Lines

**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.

**Reading Check:**

*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**:

**Reading Check:**

*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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