### Perpendicular Lines

Two lines are **perpendicular** if they meet at a **right**, angle. For a line and a point not on the line, there is exactly one line perpendicular to the line that passes through the point. There are infinitely many lines that pass through

#### Investigation: Perpendicular Line Construction; through a Point NOT on the Line

1. Draw a horizontal line and a point above that line.

Label the line

2. Take the compass and put the pointer on

3. Move the pointer to one of the arc intersections. Widen the compass a little and draw an arc below the line. Repeat this on the other side so that the two arc marks intersect.

4. Take your straightedge and draw a line from point

Notice that this is a different construction from a perpendicular bisector.

#### Investigation: Perpendicular Line Construction; through a Point on the Line

1. Draw a horizontal line and a point on that line.

Label the line

2. Take the compass and put the pointer on

3. Move the pointer to one of the arc intersections. Widen the compass a little and draw an arc above or below the line. Repeat this on the other side so that the two arc marks intersect.

4. Take your straightedge and draw a line from point

Notice that this is a different construction from a perpendicular bisector*.*

#### Perpendicular Transversals

Recall that when two lines intersect, four angles are created. If the two lines are perpendicular, then all four angles are right angles, even though only one needs to be marked with the square. Therefore, all four angles are

When a parallel line is added, then there are eight angles formed. If

Given:

Prove:

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

1. |
Given |

2. |
Definition of perpendicular lines |

3. |
Definition of a right angle |

4. |
Corresponding Angles Postulate |

5. |
Transitive PoE |

6. |
Congruent Linear Pairs |

7. |
Vertical Angles Theorem |

8. |
Definition of right angle |

9. |
Definition of perpendicular lines |

**Theorem #1:** If two lines are parallel and a third line is perpendicular to one of the parallel lines, it is also perpendicular to the other parallel line.

Or, if

**Theorem #2:** If two lines are perpendicular to the same line, they are parallel to each other.

Or, if

From these two theorems, we can now assume that any angle formed by two parallel lines and a perpendicular transversal will always be

#### Measuring Angles

1. Find

First, these two angles form a linear pair. Second, from the marking, we know that

2. Determine the measure of

From Theorem #1, we know that the lower parallel line is also perpendicular to the transversal. Therefore,

3. Find

The two adjacent angles add up to

### Examples

#### Example 1

Is

If the two adjacent angles add up to

#### Example 2

Find the value of

The two angles together make a right angle. Set up an equation and solve for

#### Example 3

Find the value of

The two angles together make a right angle. Set up an equation and solve for

### Review

Find the measure of

In questions 10-13, determine if

Find the value of

### Review (Answers)

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