# 3.8: Angles and Perpendicular Lines

- Page ID
- 4770

Lines that intersect at a 90 degree or right angle.

Two lines are **perpendicular** when they intersect to form a \(90^{\circ}\) **angle**. Below, \(l\perp \overline{AB}\).

In the definition of perpendicular the word “line” is used. However, line segments, rays and planes can also be perpendicular. The image below shows two parallel planes, with a third blue plane that is perpendicular to both of them.

## Basic Facts about Perpendicular Lines

__Theorem #1__: If \(l\parallel m\) and \(n\perp l\), then \(n\perp m\).

__Theorem #2__: If \(l\perp n\) and \(n\perp m\), then \(l\parallel m\).

__Postulate__: For any line and a point **not** on the line, there is one line perpendicular to this line passing through the point. There are infinitely many lines that pass through \(A\), but only

**that is**

*one*__perpendicular__to \(l\).

What if you were given a pair of lines that intersect each other at a \(90^{\circ}\) angle? What terminology would you use to describe such lines?

Example \(\PageIndex{1}\)

Determine the measure of \(\angle 1\).

**Solution**

We know that both parallel lines are perpendicular to the transversal.

\(m\angle 1=90^{\circ}\).

Example \(\PageIndex{2}\)

Find \(m\angle 1\).

**Solution**

The two adjacent angles add up to \(90^{\circ}\), so \(l\perp m\).

\(m\angle 1=90^{\circ}\)

because it is a vertical angle to the pair of adjacent angles and vertical angles are congruent.

Example \(\PageIndex{3}\)

Which of the following is the best example of perpendicular lines: Latitude on a Globe, Opposite Sides of a Picture Frame, Fence Posts, or Adjacent Sides of a Picture Frame?

**Solution**

The best example would be adjacent sides of a picture frame. Remember that adjacent means next to and sharing a vertex. The adjacent sides of a picture frame meet at a \(90^{\circ}\) angle and so these sides are perpendicular.

Example \(\PageIndex{4}\)

Is \(\overleftrightarrow{SO} \perp \overrightarrow{GD}\)?

**Solution**

\(\angle OGD\cong \angle SGD\) and the angles form a linear pair. This means both angles are \(90^{\circ}\), so the lines are perpendicular.

Example \(\PageIndex{5}\)

Write a 2-column proof to prove Theorem #1. * Note: You need to understand *corresponding angles

*in order to understand this proof. If you have not yet learned corresponding angles, be sure to check out that concept first, or skip this example for now.*__Given__: \(l\parallel m\), \(l\perp n\)

__Prove__: \(n\perp m\)

**Solution**

Statement |
Reason |
---|---|

1. \(l\parallel m\), \(l\perp n\) | 1. Given |

2. \(\angle 1\), \(\angle 2\), \(\angle 3\), and \(\angle 4 are right angles\) | 2. Definition of perpendicular lines |

3. \(m\angle 1=90^{\circ}\) | 3. Definition of a right angle |

4. \(m\angle 1=m\angle 5\) | 4. Corresponding Angles Postulate |

5. \(m\angle 5=90^{\circ}\) | 5. Transitive \(PoE\) |

6. \(m\angle 6=m\angle 7=90^{\circ}\) | 6. Congruent Linear Pairs |

7. \(m\angle 8=\(90^{\circ}\) | 7. Vertical Angles Theorem |

8. \(\angle 5\), \(\angle 6\), \(\angle 7\), and \(\angle 8\) are right angles | 8. Definition of right angle |

9. \(n\perp m\) | 9. Definition of perpendicular lines |

### Review

Use the figure below to answer questions 1-2. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them.

- List a pair of perpendicular lines.
- For \(\overline{AB}\), how many perpendicular lines would pass through point \(V\)? Name this/these line(s).

Use the picture below for question 3.

- If \(t\perp l\), is \(t\perp m\)? Why or why not?

Find the measure of \(\angle 1\) for each problem below.

In questions 13-16, determine if \(l\perp m.\)

Fill in the blanks in the proof below.

__Given__: \(l\perp m\), \(l\perp n\)__Prove__: \(m\parallel n\)

Statement |
Reason |
---|---|

1. | 1. |

2. \(\angle 1\) and \(\angle 2\) are right angles | 2. |

3. | 3. Definition of right angles |

4. | 4. Transitive \(PoE\) |

5. \(m\parallel n\) | 5. |

## Resources

## Vocabulary

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

perpendicular |
Two lines are when they intersect to form a \(90^{\circ}\) angle.perpendicular |

Angle |
A geometric figure formed by two rays that connect at a single point or vertex. |

Perpendicular lines |
Perpendicular lines are lines that intersect at a \(90^{\circ}\) angle. |

## Additional Resources

Interactive Element

Video: Perpendicular Lines Principles - Basic

Activities: Perpendicular Lines Discussion Questions

Study Aids: Lines and Angles Study Guide

Practice: Angles and Perpendicular Lines

Real World: Turning The Tables