Geometry (A): READ: Parallel Lines and Transversals Reading

Parallel Lines and Transversals

Learning Objectives

Identify angles formed by two parallel lines and a non-perpendicular transversal.
Identify and use the Corresponding Angles Postulate.
Identify and use the Alternate Interior Angles Theorem.
Identify and use the Alternate Exterior Angles Theorem.
Identify and use the Consecutive Interior Angles Theorem.

Introduction

In the last lesson, you learned to identify different categories of angles formed by intersecting lines. This lesson builds on that knowledge by identifying the mathematical relationships inherent within these categories.

Parallel Lines with a Transversal—Review of Terms

As a quick review, it is helpful to practice identifying different categories of angles.

Example 1

In the diagram below, two vertical parallel lines are cut by a transversal.

Identify the pairs of corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

Corresponding angles: Corresponding angles are formed on different lines, but in the same relative position to the transversal—in other words, they face the same direction. There are four pairs of corresponding angles in this diagram— $\angle{6}$ and $\angle{8}, \angle{7}$ and $\angle{1}, \angle{5}$ and $\angle{3}$ , and $\angle{4}$ and $\angle{2}$ .
Alternate interior angles: These angles are on the interior of the lines crossed by the transversal and are on opposite sides of the transversal. There are two pairs of alternate interior angles in this diagram— $\angle{7}$ and $\angle{3}$ , and $\angle{8}$ and $\angle{4}$ .
Alternate exterior angles: These are on the exterior of the lines crossed by the transversal and are on opposite sides of the transversal. There are two pairs of alternate exterior angles in this diagram— $\angle{1}$ and $\angle{5}$ , and $\angle{2}$ and $\angle{6}$ .
Consecutive interior angles: Consecutive interior angles are in the interior region of the lines crossed by the transversal, and are on the same side of the transversal. There are two pairs of consecutive interior angles in this diagram— $\angle{7}$ and $\angle{8}$ and $\angle{3}$ and $\angle{4}$ .

Corresponding Angles Postulate

By now you have had lots of practice and should be able to easily identify relationships between angles.

Corresponding Angles Postulate: If the lines crossed by a transversal are parallel, then corresponding angles will be congruent. Examine the following diagram.

You already know that $\angle{2}$ and $\angle{3}$ are corresponding angles because they are formed by two lines crossed by a transversal and have the same relative placement next to the transversal. The Corresponding Angles postulate says that because the lines are parallel to each other, the corresponding angles will be congruent.

Example 2

In the diagram below, lines $p$ and $q$ are parallel. What is the measure of $\angle{1}$ ?

Because lines $p$ and $q$ are parallel, the $120^\circ$ angle and $\angle{1}$ are corresponding angles, we know by the Corresponding Angles Postulate that they are congruent. Therefore, $m\angle{1} = 120^\circ$ .

Alternate Interior Angles Theorem

Now that you know the Corresponding Angles Postulate, you can use it to derive the relationships between all other angles formed when two lines are crossed by a transversal. Examine the angles formed below.

If you know that the measure of $\angle{1}$ is $120^\circ,$ you can find the measurement of all the other angles. For example, $\angle{1}$ and $\angle{2}$ must be supplementary (sum to $180^\circ$ ) because together they are a linear pair (we are using the Linear Pair Postulate here). So, to find $m\angle{2}$ , subtract $120^\circ$ from $180^\circ.$

$m \angle {2} & = 180^\circ - 120^\circ \\ m \angle {2} & = 60^\circ$

So, $m\angle{2}=60^\circ$ . Knowing that $\angle{2}$ and $\angle{3}$ are also supplementary means that $m \angle {3}=120^\circ,$ since $120+60=180$ . If $m\angle{3}=120^\circ$ , then $m \angle {4}$ must be $60^\circ,$ because $\angle{3}$ and $\angle{4}$ are also supplementary. Notice that $\angle{1} \cong \angle{3}$ (they both measure $120^\circ$ ) and $\angle{2} \cong \angle{4}$ (both measure $60^\circ$ ). These angles are called vertical angles. Vertical angles are on opposite sides of intersecting lines, and will always be congruent by the Vertical Angles Theorem, which we proved in an earlier chapter. Using this information, you can now deduce the relationship between alternate interior angles.

Example 3

Lines $l$ and $m$ in the diagram below are parallel. What are the measures of angles $\alpha$ and $\beta$ ?

In this problem, you need to find the angle measures of two alternate interior angles given an exterior angle. Use what you know. There is one angle that measures $80^\circ.$ Angle $\beta$ corresponds to the $80^\circ$ angle. So by the Corresponding Angles Postulate, $m \angle{\beta} = 80^\circ$ .

Now, because $\angle{\alpha}$ is made by the same intersecting lines and is opposite the $80^\circ$ angle, these two angles are vertical angles. Since you already learned that vertical angles are congruent, we conclude $m \angle{\alpha} = 80^\circ$ . Finally, compare angles $\alpha$ and $\beta$ . They both measure $80^\circ,$ so they are congruent. This will be true any time two parallel lines are cut by a transversal.

We have shown that alternate interior angles are congruent in this example. Now we need to show that it is always true for any angles.

Alternate Interior Angles Theorem

Alternate interior angles formed by two parallel lines and a transversal will always be congruent.

Given: $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel lines crossed by transversal $\overleftrightarrow{XY}$
Prove that Alternate Interior Angles are congruent

Note: It is sufficient to prove that one pair of alternate interior angles are congruent. Let's focus on proving $\angle{DWZ} \cong \angle{WZA}$ .

Statement	Reason
1. $\overleftrightarrow{AB} \\| \overleftrightarrow{CD}$	1.Given
2. $\angle{DWZ} \cong \angle{BZX}$	2. Corresponding Angles Postulate
3. $\angle{BZX} \cong \angle{WZA}$	3. Vertical Angles Theorem
4. $\angle{DWZ} \cong \angle{WZA}$	4. Transitive property of congruence

Alternate Exterior Angles Theorem

Now you know that pairs of corresponding, vertical, and alternate interior angles are congruent. We will use logic to show that Alternate Exterior Angles are congruent—when two parallel lines are crossed by a transversal, of course.

Example 4

Lines $g$ and $h$ in the diagram below are parallel. If $m \angle{4} = 43^\circ$ , what is the measure of $\angle{5}$ ?

You know from the problem that $m \angle{4} = 43^\circ$ . That means that $\angle{4}'s$ corresponding angle, which is $\angle{3}$ , will measure $43^\circ$ as well.

The corresponding angle you just filled in is also vertical to $\angle{5}$ . Since vertical angles are congruent, you can conclude $m \angle{5} = 43^\circ$ .

This example is very similar to the proof of the alternate exterior angles Theorem. Here we write out the theorem in whole:

Alternate Exterior Angles Theorem

If two parallel lines are crossed by a transversal, then alternate exterior angles are congruent.

We omit the proof here, but note that you can prove alternate exterior angles are congruent by following the method of example 4, but not using any particular measures for the angles.

Consecutive Interior Angles Theorem

The last category of angles to explore in this lesson is consecutive interior angles. They fall on the interior of the parallel lines and are on the same side of the transversal. Use your knowledge of corresponding angles to identify their mathematical relationship.

Example 5

Lines $r$ and $s$ in the diagram below are parallel. If the angle corresponding to $\angle{1}$ measures $76^\circ,$ what is $m\angle{2}$ ?

This process should now seem familiar. The given $76^\circ$ angle is adjacent to $\angle{2}$ and they form a linear pair. Therefore, the angles are supplementary. So, to find $m\angle{2}$ , subtract $76^\circ$ from $180^\circ.$

$m\angle{2} & = 180 -76\\ m\angle{2} & = 104^\circ$

This example shows that if two parallel lines are cut by a transversal, the consecutive interior angles are supplementary; they sum to $180^\circ.$ This is called the Consecutive Interior Angles Theorem. We restate it here for clarity.

Consecutive Interior Angles Theorem

If two parallel lines are crossed by a transversal, then consecutive interior angles are supplementary.

Proof: You will prove this as part of your exercises.

Multimedia Link Now that you know all these theorems about parallel lines and transverals, it is time to practice. In the following game you use apply what you have learned to name and describe angles formed by a transversal. Interactive Angles Game.

Lesson Summary

In this lesson, we explored how to work with different angles created by two parallel lines and a transversal. Specifically, we have learned:

How to identify angles formed by two parallel lines and a non-perpendicular transversal.
How to identify and use the Corresponding Angles Postulate.
How to identify and use the Alternate Interior Angles Theorem.
How to identify and use the Alternate Exterior Angles Theorem.
How to identify and use the Consecutive Interior Angles Theorem.

These will help you solve many different types of problems. Always be on the lookout for new and interesting ways to analyze lines and angles in mathematical situations.

Points To Consider

You used logic to work through a number of different scenarios in this lesson. Always apply logic to mathematical situations to make sure that they are reasonable. Even if it doesn’t help you solve the problem, it will help you notice careless errors or other mistakes.

The following questions are for your own benefit. The answers are listed below the questions in order for you to check your work.

Review Questions

Solve each problem.

Use the diagram below for Questions 1-4. In the diagram, lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel.

What term best describes the relationship between and
1. alternate exterior angles
2. consecutive interior angles
3. corresponding angles
4. alternate interior angles
What term best describes the mathematical relationship between and ?
1. congruent
2. supplementary
3. complementary
4. no relationship
What term best describes the relationship between and
1. alternate exterior angles
2. consecutive interior angles
3. complementary
4. alternate interior angles
What term best describes the mathematical relationship between and
1. congruent
2. supplementary
3. complementary
4. no relationship
Use the diagram below for questions 5-7. In the diagram, lines $l$ and $m$ are parallel $\gamma , \beta , \theta$ represent the measures of the angles.
What is $\gamma?$
What is $\beta?$
What is $\theta?$

The map below shows some of the streets in Ahmed’s town.

Jimenez Ave and Ella Street are parallel. Use this map to answer questions 8-10.

What is the measure of angle 1?
What is the measure of angle 2?
What is the measure of angle 3?
Prove the Consecutive Interior Angle Theorem. Given $r || s$ , prove $\angle{1}$ and $\angle{2}$ are supplementary.

Review Answers

c
a
d
b
$73^\circ$
$107^\circ$
$107^\circ$
$65^\circ$
$65^\circ$
$115^\circ$

Proof of Consecutive Interior Angle Theorem. Given $r || s$ , prove $\angle{1}$ and $\angle{2}$ are supplementary.

Statement	Reason
1. $r \|\| s$	1. Given
2. $\angle{1} \cong \angle{3}$	2. Corresponding Angles Postulate
3. $\angle{2}$ and $\angle{3}$ are supplementary	3. Linear Pair Postulate
4. $m\angle{2} + m\angle{3}=180^\circ$	4. Definition of supplementary angles
5. $m\angle{2} + m\angle{1}=180^\circ$	5. Substitution $(\angle{1} \cong \angle{3})$
6. $\angle{2}$ and $\angle{1}$ are supplementary	6. Definition of supplementary angles

Last modified: Monday, June 28, 2010, 1:09 PM