# 3.4: Properties of Perpendicular Lines

**At Grade**Created by: CK-12

## Learning Objectives

- Understand the properties of perpendicular lines.

## Review Queue

- Draw a picture of two parallel lines,
l andm , and a transversal that is perpendicular tol . Is the transversal perpendicular tom too? -
m∠A=35∘ and is complementary to∠B . What ism∠B ? -
m∠C=63∘ and is complementary to∠D . What ism∠D ? - Draw a picture of a linear pair where the angles are congruent. What is the measure of each angle?

**Know What?** There are several examples of slope in nature. To the right are pictures of Half Dome, in Yosemite National Park and the horizon over the Pacific Ocean. These are examples of horizontal and vertical lines in real life. What is the slope of these lines?

## Congruent Linear Pairs

A linear pair is a pair of adjacent angles whose outer sides form a straight line. The Linear Pair Postulate says that the angles in a linear pair add up to

*If a linear pair is congruent, the angles are both 90∘.*

**Example 1:** Determine the measure of

**Solution:** The angles are congruent and form a linear pair.

Both angles are

**Example 2:** Find

**Solution:** These two angles form a linear pair and

## Perpendicular Transversals

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*. All four angles are also

If

**Example 3:** Write a 2-column proof.

Given:

Prove:

**Solution:**

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

1. |
Given |

2. |
Definition of perpendicular lines |

3. |
Definition of a right angle |

4. |
Corresponding Angles Postulate |

5. |
Transitive |

6. |
Congruent Linear Pairs |

7. |
Vertical Angles Theorem |

8. |
Definition of right angle |

9. |
Definition of perpendicular lines |

**Theorem 3-1:** If

**Theorem 3-2:** If

*Every angle in the above two theorems will always be*

**Example 4:** Determine the measure of

**Solution:** From Theorem 3-1, we know that both parallel lines are perpendicular to the transversal.

**Adjacent Complementary Angles** If complementary angles are adjacent, their nonadjacent sides are perpendicular rays. What you have learned about perpendicular lines can be applied here.

**Example 5:** Find

**Solution:** The two adjacent angles add up to

**Example 6:** What is the measure of \begin{align*}\angle 1\end{align*}?

**Solution:** \begin{align*}l \perp m\end{align*}

\begin{align*}\text{So}, \ m\angle 1 + 70^\circ & = 90^\circ\\ m\angle 1 & = 20^\circ\end{align*}

**Example 7:** Is \begin{align*}l \perp m\end{align*}? Explain why or why not.

**Solution:** If the two adjacent angles add up to \begin{align*}90^\circ\end{align*}, then \begin{align*}l\end{align*} and \begin{align*}m\end{align*} are perpendicular.

\begin{align*}23^\circ + 67^\circ = 90^\circ\end{align*}

Therefore, \begin{align*}l \perp m\end{align*}.

**Know What? Revisited**

Half Dome is vertical, so the slope is undefined.

http://www.nps.gov/yose/index.htm

Any horizon over an ocean is horizontal, which has a slope of zero, or no slope.

If Half Dome was placed on top of the ocean, the two would be perpendicular.

## Review Questions

- Questions 1-25 are similar to Examples 1, 2, 4, 5, 6 and 7.
- Question 26 is similar to Example 3.
- Questions 27-30 are similar to Examples 5 and 6 and use Algebra.

Find the measure of \begin{align*}\angle 1\end{align*} for each problem below.

For questions 10-13, use the picture below.

- Find \begin{align*}m\angle ACD\end{align*}.
- Find \begin{align*}m\angle CDB\end{align*}.
- Find \begin{align*}m\angle EDB\end{align*}.
- Find \begin{align*}m\angle CDE\end{align*}.

In questions 14-17, determine if \begin{align*}l \perp m\end{align*}.

For questions 18-25, use the picture below.

- Find \begin{align*}m\angle 1\end{align*}.
- Find \begin{align*}m\angle 2\end{align*}.
- Find \begin{align*}m\angle 3\end{align*}.
- Find \begin{align*}m\angle 4\end{align*}.
- Find \begin{align*}m\angle 5\end{align*}.
- Find \begin{align*}m\angle 6\end{align*}.
- Find \begin{align*}m\angle 7\end{align*}.
- Find \begin{align*}m\angle 8\end{align*}.

Fill in the blanks in the proof below.

- Given: \begin{align*}l \perp m, \ l \perp n\end{align*} Prove: \begin{align*}m || n\end{align*}

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

1. | |

2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are right angles | |

3. | Definition of right angles |

4. | Transitive \begin{align*}PoE\end{align*} |

5. \begin{align*}m || n\end{align*} |

** Algebra Connection** Find the value of \begin{align*}x\end{align*}.

## Review Queue Answers

- Yes, the transversal will be perpendicular to \begin{align*}m\end{align*}.
- \begin{align*}m\angle B = 55^\circ\end{align*}
- \begin{align*}m\angle D = 27^\circ\end{align*}
- Each angle is \begin{align*}90^\circ\end{align*}.

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