2.6: Parallel Lines and Transversals: Identifying Angle Pairs, Part 2
Learning Objectives
 Identify angles formed by two parallel lines and a nonperpendicular transversal.
 Identify and use the Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, and SameSide Interior Angles Theorem.
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: and , ____________, ____________, and ____________.

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 — and and and and and .
(Using the same diagram on the previous page):
Alternate interior angles: and , and ____________.
 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 — and and and .
Alternate exterior angles: and and ____________.
 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 — and and and .
Consecutive (sameside) interior angles: and and ____________.
 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 — and and and .
Angle Postulates and Theorems
By now you have had lots of practice and should be able to identify relationships between angles.
Do you remember the difference between a postulate and a theorem?
The difference is...
A postulate does not have to be proven. It is selfevident, or obvious. A theorem is not obvious. It has to be proven.
A postulate does not have to be ______________________________.
A ______________________________ must be proven.
We will explore a number of postulates and theorems that involve the different types of angle relationships you just learned.
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 and 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 (which we can tell because of the similar arrows on them), the corresponding angles will be congruent.
 We know the two lines in the diagram above are parallel because they are both marked with __________________________.
 The Corresponding Angles Postulate tells us that corresponding angles are __________________________________ if the transversal crosses parallel lines.
Reading Check:
In the diagram below, lines and are parallel. What is the measure of ?
Because lines and are parallel, the angle and are corresponding angles. We know by the Corresponding Angles Postulate that they are congruent. Therefore, .
Alternate Interior Angles Theorem
You can use the Corresponding Angles Postulate to derive the relationships between all other angles formed when two lines are crossed by a transversal.
Look at the diagram below and the chart on the following page:
If you know that the measure of angle 1 is , you can find the measures of the other angles in the picture.
What are the measures of angles 2, 3, and 4?
Try to figure them out and then check your work in the chart on the next page.
Angle Measure  How We Know 

Because it makes a linear pair with . Remember, linear pairs are supplementary. That means they add to be


There are two ways to find the measure of . First, and are vertical angles, and vertical angles are congruent.
Or, you can find the measure of Angle 3 using linear pairs. and make a linear pair. This means they are supplementary (add to ). We already know that , so must be .


There are two ways to find the measure of . First, and are vertical angles, and vertical angles are congruent. We know that
Or, you can find the measure of Angle 4 using linear pairs. and make a linear pair. So do and . This means they are supplementary (add to ). We already know that , so must be .

As you can see from this chart, there are some very important relationships we already know that can help us find the measure of missing angles! In this example, the ones we used multiple times are:
 Linear pairs are supplementary and
 Vertical angles are congruent
Example 2
Lines and in the diagram below are parallel. What are the measures of angles and ?
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 . Angle and the angle are corresponding angles. So by the Corresponding Angles Postulate, .
 Now, because is made by the same intersecting lines and is opposite the angle, these two angles are vertical angles. Since you already learned that vertical angles are congruent, we conclude .
 Finally, compare angles and . They both measure , so they are congruent. This will be true any time two parallel lines are cut by a transversal.
Alternate Interior Angles Theorem
Alternate interior angles formed by two parallel lines and a transversal are always congruent.
 When two parallel lines are crossed by a transversal, the alternate interior angles are _________________________________.
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 3
Lines and in the diagram below are parallel. If , what is the measure of ?
You know from the problem that . That means that ’s corresponding angle, which is , will measure as well:
The corresponding angle you just filled in is also vertical to . Since vertical angles are congruent, you can conclude that :
So, is congruent to . In other words, the alternate exterior angles are congruent.
Alternate Exterior Angles Theorem
If two parallel lines are crossed by a transversal, then alternate exterior angles are congruent.
In Example 3 on the previous page, we proved the Alternate Exterior Angles Theorem. We figured this out because:
 When lines are parallel, corresponding angles are ___________________________.
 Then, all vertical angles are _______________________________.
These led us to our conclusion that:
 When two parallel lines are cut by a transversal, the alternate exterior angles formed are _______________________________.
Consecutive (SameSide) 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 4
Lines and in the diagram below are parallel.
If the angle corresponding to measures , what is ?
This process should now seem familiar. The given angle is adjacent to and they form a linear pair. Therefore, the angles are supplementary.
Since supplementary angles add up to __________, find by subtracting from :
Example 4 on the last page shows that if two parallel lines are cut by a transversal, the consecutive interior angles are supplementary; they sum to .
This is called the Consecutive Interior Angles Theorem. We restate it below for clarity.
Consecutive (SameSide) Interior Angles Theorem
If two parallel lines are crossed by a transversal, then consecutive interior angles are supplementary.
 Consecutive Interior Angles and SameSide Interior Angles are the ______________.
 When two parallel lines are crossed by a transversal, sameside interior angles are _______________________________________.
Reading Check:
In the diagram below, lines and are being cut by transversal .
For questions 13, circle the best answer.
1. Which postulate or theorem explains why angles and are congruent?
a. Corresponding Angles Postulate
b. Alternate Interior Angles Theorem
c. Alternate Exterior Angles Theorem
d. Consecutive (SameSide) Interior Angles Theorem
2. If , what will be the measure of ? What is the reason for your answer?
a. ; Corresponding Angles Postulate
b. ; Alternate Exterior Angles Theorem
c. ; Corresponding Angles Postulate
d. ; Alternate Exterior Angles Theorem
3. Which postulate or theorem states that an angle pair is supplementary (not congruent)?
a. Corresponding Angles Postulate
b. Alternate Interior Angles Theorem
c. Alternate Exterior Angles Theorem
d. Consecutive (SameSide) Interior Angles Theorem
4. Label all of the angle measures in the diagram above.
Graphic Organizer for Lesson 3
Angle Pairs  Congruent or Supplementary?  Picture  

Corresponding  
Alternate  Interior  
Alternate  Exterior  
Sameside/Consecutive Interior 
Image Attributions
Description
Authors:
Tags:
Concept Nodes:
Date Created:
Feb 23, 2012Last Modified:
May 12, 2014If you would like to associate files with this None, please make a copy first.