### Perpendicular Lines

Two lines are **perpendicular** when they intersect to form a

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

Theorem #2: If

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

**that is perpendicular to**

*one*

What if you were given a pair of lines that intersect each other at a

### Examples

#### Example 1

Determine the measure of

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

#### Example 2

Find

The two adjacent angles add up to

#### Example 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?

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

#### Example 4

Is

#### Example 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:

Prove:

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

1. |
1. Given |

2. |
2. Definition of perpendicular lines |

3. |
3. Definition of a right angle |

4. |
4. Corresponding Angles Postulate |

5. |
5. Transitive |

6. |
6. Congruent Linear Pairs |

7. |
7. Vertical Angles Theorem |

8. |
8. Definition of right angle |

9. |
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
AB¯¯¯¯¯¯¯¯ , how many perpendicular lines would pass through pointV ? Name this/these line(s).

Use the picture below for question 3.

- If
t⊥l , ist⊥m ? Why or why not?

Find the measure of

In questions 13-16, determine if

Fill in the blanks in the proof below.

- Given:
l⊥m, l⊥n Prove:m||n

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

1. | 1. |

2. |
2. |

3. | 3. Definition of right angles |

4. | 4. Transitive |

5. |
5. |

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.2.

### Resources