3.2: Properties of Parallel Lines
Learning Objectives
- Use the Corresponding Angles Postulate.
- Use the Alternate Interior Angles Theorem.
- Use the Alternate Exterior Angles Theorem.
- Use Same Side Interior Angles Theorem.
Review Queue
Use the picture below to determine:
- A pair of corresponding angles.
- A pair of alternate interior angles.
- A pair of same side interior angles.
- If , what other angles do you know?
Know What? The streets below are in Washington DC. The red street is R St. and the blue street is Q St. These two streets are parallel. The transversals are: Rhode Island Ave. (green) and Florida Ave. (orange).
- If , determine the other angles that are .
- If , determine the other angles that are .
- Why do you think the “State Streets” exists? Why aren’t all the streets parallel or perpendicular?
In this section, we are going to discuss a specific case of two lines cut by a transversal. The two lines are now going to be parallel. If the two lines are parallel, all of the angles, corresponding, alternate interior, alternate exterior and same side interior have new properties. We will begin with corresponding angles.
Corresponding Angles Postulate
Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
If and both are cut by , then , and .
must be parallel to in order to use this postulate. Recall that a postulate is just like a theorem, but does not need to be proven. We can take it as true and use it just like a theorem from this point.
Investigation 3-4: Corresponding Angles Exploration
You will need: paper, ruler, protractor
- Place your ruler on the paper. On either side of the ruler, draw lines, 3 inches long. This is the easiest way to ensure that the lines are parallel.
- Remove the ruler and draw a transversal. Label the eight angles as shown.
- Using your protractor, measure all of the angles. What do you notice?
In this investigation, you should see that and . by the Vertical Angles Theorem. By the Corresponding Angles Postulate, we can say and therefore by the Transitive Property. You can use this reasoning for the other set of congruent angles as well.
Example 1: If , what is ?
Solution: and are corresponding angles and , from the markings in the picture. By the Corresponding Angles Postulate the two angles are equal, so .
Example 2: Using the measures of and from Example 2, find all the other angle measures.
Solution: If , then because they are a linear pair. is a vertical angle with , so . and are vertical angles, so . By the Corresponding Angles Postulate, we know , and , so , and .
Alternate Interior Angles Theorem
Example 3: Find .
Solution: because they are corresponding angles and the lines are parallel. and are vertical angles, so also.
and the angle are alternate interior angles.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
Proof of Alternate Interior Angles Theorem
Given:
Prove:
Statement | Reason |
---|---|
1. | Given |
2. | Corresponding Angles Postulate |
3. | Vertical Angles Theorem |
4. | Transitive PoC |
There are several ways we could have done this proof. For example, Step 2 could have been for the same reason, followed by . We could have also proved that .
Example 4: Algebra Connection Find the measure of the angle and .
Solution: The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and solve for .
Alternate Exterior Angles Theorem
Example 5: Find and .
Solution: because they are vertical angles. Because the lines are parallel, by the Corresponding Angles Theorem. Therefore, .
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.
The proof of this theorem is very similar to that of the Alternate Interior Angles Theorem and you will be asked to do in the exercises at the end of this section.
Example 6: Algebra Connection Find the measure of each angle and the value of .
Solution: The given angles are alternate exterior angles. Because the lines are parallel, we can set the expressions equal to each other to solve the problem.
If , then each angle is , or .
Same Side Interior Angles Theorem
Same side interior angles have a different relationship that the previously discussed angle pairs.
Example 7: Find .
Solution: Here, because they are alternate interior angles. and are a linear pair, so they are supplementary.
This example shows that if two parallel lines are cut by a transversal, the same side interior angles are supplementary.
Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.
If and both are cut by , then
and .
You will be asked to do the proof of this theorem in the review questions.
Example 8: Algebra Connection Find the measure of .
Solution: The given angles are same side interior angles. The lines are parallel, therefore the angles add up to . Write an equation.
While you might notice other angle relationships, there are no more theorems to worry about. However, we will continue to explore these other angle relationships. For example, same side exterior angles are also supplementary. You will prove this in the review questions.
Example 9: and . Prove .
Solution:
Statement | Reason |
---|---|
1. and | Given |
2. | Corresponding Angles Postulate |
3. | Alternate Exterior Angles Theorem |
4. | Transitive PoC |
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 . You could argue that the “State Streets” exist to help traffic move faster and more efficiently through the city.
Review Questions
For questions 1-7, determine if each angle pair below is congruent, supplementary or neither.
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- and
- and
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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.
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- List all angles congruent to .
For 17-20, find the values of and .
Algebra Connection For questions 21-25, use the picture below. Find the value of and/or .
- Fill in the blanks in the proof 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 |
For 27 and 28, use the picture to the right to complete each proof.
- Given: Prove: (Alternate Exterior Angles Theorem)
- Given: Prove: and are supplementary
For 29-31, use the picture to the right to complete each proof.
- Given: Prove:
- Given: Prove:
- Given: Prove: and are supplementary
- Find the measures of all the numbered angles in the figure below.
Algebra Connection For 33 and 34, find the values of and .
- Error Analysis Nadia is working on Problem 31. Here is her proof:
Statement | Reason |
---|---|
1. | Given |
2. | Alternate Exterior Angles Theorem |
3. | Same Side Interior Angles Theorem |
4. | Vertical Angles Theorem |
5. | Transitive PoC |
What happened? Explain what is needed to be done to make the proof correct.
Review Queue Answers
- and and and , or and
- and or and
- and or and
- and or and