# 3.2: Properties of Parallel Lines

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

## Learning Objectives

- Determine what happens to corresponding angles, alternate interior angles, alternate exterior angles, and same side interior angles when two lines are parallel.

## Review Queue

Use the picture below to determine:

- A pair of corresponding angles.
- A pair of alternate interior angles.
- A pair of alternate exterior angles.
- A pair of same side interior angles.

**Know What?** The streets below are in Washington DC. The red street and the blue street are parallel. The transversals are the green and orange streets.

- If , determine the other angles that are .
- If , determine the other angles that are .

## Corresponding Angles Postulate

**Corresponding Angles Postulate:** If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

If , then .

**Example 1:** If , which pairs of angles are congruent by the Corresponding Angles Postulate?

**Solution:** There are 4 pairs of congruent corresponding angles:

, and .

**Investigation 3-4: Corresponding Angles Exploration**

1. Place your ruler on the paper. On either side of the ruler, draw 2 lines, 3 inches long.

This is the easiest way to ensure that the lines are parallel.

2. Remove the ruler and draw a transversal. Label the eight angles as shown.

3. Using your protractor, measure all of the angles. What do you notice?

**You should notice that all the corresponding angles have equal measures.**

**Example 2:** If , what is ?

**Solution:** and are corresponding angles and from the arrows on them. by the Corresponding Angles Postulate, which means that .

**Example 3:** Using the measures of and from Example 2, find all the other angle measures.

**Solution:** If , then (linear pair). (vertical angles), so (vertical angle with ).

By the Corresponding Angles Postulate, we know , and , so , and .

## Alternate Interior Angles Theorem

**Example 4:** Find .

**Solution:** because they are corresponding angles and the lines are parallel. and are vertical angles, so .

**Alternate Interior Angles Theorem:** If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

If , then

**Proof of Alternate Interior Angles Theorem**

Given:

Prove:

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

1. | Given |

2. | Corresponding Angles Postulate |

3. | Vertical Angles Theorem |

4. | Transitive |

We could have also proved that .

**Example 5:** ** Algebra Connection** Find the measure of .

**Solution:** The two given angles are alternate interior angles and equal.

**Alternate Exterior Angles Theorem**

**Example 6:** Find and .

**Solution:** by vertical angles. The lines are parallel, so by the Corresponding Angles Postulate.

Here, and are alternate exterior angles.

**Alternate Exterior Angles Theorem:** If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

If , then .

**Example 7:** ** Algebra Connection** Find the measure of each angle and the value of .

**Solution:** The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.

If , then each angle is .

**Same Side Interior Angles Theorem**

Same side interior angles are on the interior of the parallel lines and on the same side of the transversal. They have a different relationship that the other angle pairs.

**Example 8:** Find .

**Solution:** and are alternate interior angles, so . and are a linear pair, so they add up to .

This example shows that if two parallel lines are cut by a transversal, the same side interior angles add up to .

**Same Side Interior Angles Theorem:** If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.

If , then .

**Example 9:** Find , and .

**Solution:** by Alternate Interior Angles

because it is a linear pair with .

by Same Side Interior Angles.

**Example 10:** ** Algebra Connection** Find the measure of .

**Solution:** The given angles are same side interior angles. Because the lines are parallel, the angles add up to .

**Example 11:** and . Explain how .

**Solution:** Because and are not on the same transversal, we cannot assume they are congruent.

by Corresponding Angles

by Alternate Exterior Angles

by the Transitive Property

**Know What? Revisited** Using what we have learned in this lesson, the other angles that are are , and the vertical angle with . The other angles that are are , and the vertical angle with .

## Review Questions

- Questions 1-7 use the theorems learned in this section.
- Questions 8-16 are similar to Example 11.
- Questions 17-20 are similar to Example 6, 8 and 9.
- Questions 21-25 are similar to Examples 5, 7, and 10.
- Questions 26-29 are similar to the proof of the Alternate Interior Angles Theorem.
- Question 30 uses the theorems learned in this section.

For questions 1-7, determine if each angle pair below is congruent, supplementary or neither.

- and
- and
- and
- and
- and
- and
- and

For questions 8-16, determine if the angle pairs below are: Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Same Side Interior Angles, Vertical Angles, Linear Pair or None.

- and
- and
- and
- and
- and
- and
- and
- and
- List all angles congruent to .

For 17-20, find the values of and .

** Algebra Connection** For questions 21-25, use the picture to the right. Find the value of and/or .

Fill in the blanks in the proofs below.

- Given: Prove: and are supplementary (Same Side Interior Angles Theorem)

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

1. | Given |

2. | |

3. | angles have = measures |

4. | Linear Pair Postulate |

5. | Definition of Supplementary Angles |

6. | |

7. and are supplementary |

- Given: Prove: (Alternate Exterior Angles Theorem)

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

1. | |

2. | |

3. | Vertical Angles Theorem |

4. |

For 28 and 29, use the picture to the right.

- Given: Prove:

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

1. | |

2. | |

3. | Corresponding Angles Postulate |

4. |

- Given: Prove: and are supplementary

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

1. | |

2. | |

3. | |

4. | Same Side Interior Angles |

5. an are supplementary |

- Find the measures of all the numbered angles in the figure below.

## Review Queue Answers

- and and and , or and
- and or and
- and or and
- and or and