CK-12 Geometry
Page 10
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: and are parallel lines crossed by transversal
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 .
Statement Reason
1. 1.Given
2. 2. Corresponding Angles Postulate
3. 3. Vertical Angles Theorem
4. 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 and in the diagram below are parallel. If , what is the measure of ?
You know from the problem that . That means that 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 .
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 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. So, to find , subtract from
This example 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 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 error.pngs or other mistakes.
Review Questions
Solve each problem.
Use the diagram below for Questions 1-4. In the diagram, lines and are parallel.
What term best describes the relationship between and alternate exterior angles
consecutive interior angles
corresponding angles
alternate interior angles
What term best describes the mathematical relationship between and ? congruent
supplementary
complementary
no relationship
What term best describes the relationship between and alternate exterior angles
consecutive interior angles
complementary
alternate interior angles
What term best describes the mathematical relationship between and congruent
supplementary
complementary
no relationship
Use the diagram below for questions 5-7. In the diagram, lines and are parallel represent the measures of the angles.
What is
What is
What is
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 , prove and are supplementary.
Review Answers
c
a
d
b
Proof of Consecutive Interior Angle Theorem. Given , prove and are supplementary.
Statement Reason
1. 1. Given
2. 2. Corresponding Angles Postulate
3. and are supplementary 3. Linear Pair Postulate
4. 4. Definition of supplementary angles
5. 5. Substitution
6. and are supplementary 6. Definition of supplementary angles
Proving Lines Parallel
Learning Objectives
Identify and use the Converse of the Corresponding Angles Postulate.
Identify and use the Converse of Alternate Interior Angles Theorem.
Identify and use the Converse of Alternate Exterior Angles Theorem.
Identify and use the Converse of Consecutive Interior Angles Theorem.
Identify and use the Parallel Lines Property.
Introduction
If two angles are vertical angles, then they are congruent. You learned this as the Vertical Angles Theorem. Can you reverse this statement? Can you swap the “if” and “then” parts and will the statement still be true?
The converse of a logical statement is made by reversing the hypothesis and the conclusion in an if-then statement. With the Vertical Angles Theorem, the converse is “If two angles are congruent then they are vertical angles.” Is that a true statement? In this case, no. The converse of the Vertical Angles Theorem is NOT true. There are many examples of congruent angles that are not vertical angles—for example the corners of a square.
Sometimes the converse of an if-then statement will also be true. Can you think of an example of a statement in which the converse is true? This lesson explores converses to the postulates and theorems about parallel lines and transversals.
Corresponding Angles Converse
Let’s apply the concept of a converse to the Corresponding Angles Postulate. Previously you learned that "if two parallel lines are cut by a transversal, the corresponding angles will be congruent." The converse of this statement is "if corresponding angles are congruent when two lines are cut by a transversal, then the two lines crossed by the transversal are parallel." This converse is true, and it is a postulate.
Converse of Corresponding Angles Postulate
If corresponding angles are congruent w
hen two lines are crossed by a transversal, then the two lines crossed by the transversal are parallel.
Example 1
Suppose we know that and . What can we conclude about lines and ?
Notice that and are corresponding angles. Since , we can apply the Converse of the Corresponding Angles Postulate and conclude that .
You can also use converse statements in combination with more complex logical reasoning to prove whether lines are parallel in real life contexts. The following example shows a use of the contrapositive of the Corresponding Angles Postulate.
Example 2
The three lines in the figure below represent metal bars and a cable supporting a water tower.
and . Are the lines and parallel?
To find out whether lines and are parallel, you must identify the corresponding angles and see if they are congruent. In this diagram, and are corresponding angles because they are formed by the transversal and the two lines crossed by the transversal and they are in the same relative place.
The problem states that and . Thus, they are not congruent. If those two angles are not congruent, the lines are not parallel. In this scenario, the lines and (and thus the support bars they represent) are NOT parallel.
Note that just because two lines may look parallel in the picture that is not enough information to say that the lines are parallel. To prove two lines are parallel you need to look at the angles formed by a transversal.
Alternate Interior Angles Converse
Another important theorem you derived in the last lesson was that when parallel lines are cut by a transversal, the alternate interior angles formed will be congruent. The converse of this theorem is, “If alternate interior angles formed by two lines crossed by a transversal are congruent, then the lines are parallel.” This statement is also true, and it can be proven using the Converse of the Corresponding Angles Postulate.
Converse of Alternate Interior Angles Theorem
If two lines are crossed by a transversal and alternate interior angles are congruent, then the lines are parallel.
Given and are crossed by and .
Prove
Statement Reason
1. and are crossed by and . 1. Given
2. 2. Vertical Angles Theorem
3. 3. Transitive Property of Angle Congruence
4. 4. Converse of the Corresponding Angles Postulate.
Notice in the proof that we had to show that the corresponding angles were congruent. Once we had done that, we satisfied the conditions of the Converse of the Corresponding Angles postulate, and we could use that in the final step to prove that the lines are parallel.
Example 3
Are the two lines in this figure parallel?
This figure shows two lines that are cut by a transversal. We don't know . However, if you look at its linear pair, that angle has a measure of By the Linear Pair Postulate, this angle is supplementary to . In other words, the sum of and will be Use subtraction to find .
So, . Now look and . is a vertical angle with the angle measuring By the Vertical Angles Theorem, .
Since as can apply the converse of the Alternate Interior Angles Theorem to conclude that .
Notice in this example that you could have also used the Converse of the Corresponding Angles Postulate to prove the two lines are parallel. Also, This example highlights how, if a figure is not drawn to scale you cannot assume properties of the objects in the figure based on looks.
Converse of Alternate Exterior Angles
The more you practice using the converse of theorems to find solutions, the easier it will become. You have already probably guessed that the converse of the Alternate Exterior Angles Theorem is true.
Converse of the Alternate Exterior Angles Theorem
If two lines are crossed by a transversal and the alternate exterior angles are congruent, then the lines crossed by the transversal are parallel.
Putting together the alternate exterior angles theorem and its converse, we get the biconditional statement: Two lines crossed by a transversal are parallel if and only if alternate exterior angles are congruent.
Use the example below to apply this concept to a real-world situation.
Example 4
The map below shows three roads in Julio’s town.
In Julio's town, Franklin Way and Chavez Avenue are both crossed by Via La Playa. Julio used a surveying tool to measure two angles in the intersections as shown and he drew the sketch above (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue. How can he solve this problem and what is the correct answer?
Notice that this question asks you not only to identify the answer, but also the process required to solve it. Make sure that your solution is step-by-step so that anyone reading it can follow your logic.
To begin, notice that the labeled angle and are alternate exterior angles. If these two angles are congruent, then the lines are parallel. If they are not congruent, the lines are not parallel. To find the measure of angle , you can use the other angle labeled in this diagram, measuring This angle is supplementary to because they are a linear pair. Using the knowledge that a linear pair must be supplementary, find the value of .
Angle is equal to This angle is wider than the other alternate exterior angle, which measures so the alternate exterior angles are not congruent. Therefore, Franklin Way and Chavez Avenue are not parallel streets.
In this example, we used the contrapositive of the converse of the Alternate Exterior Angles Theorem to prove that the two lines were not parallel.
Converse of Consecutive Interior Angles
The final converse theorem to explore in this lesson addressed the Consecutive Interior Angles Theorem. Remember that these angles aren’t congruent when lines are parallel, they are supplementary. In other words, if the two lines are parallel, the angles on the interior and on the same side of the transversal will sum to So, if two consecutive interior angles made by two lines and a transversal add up to the two lines that form the consecutive angles are parallel.
Example 5
Identify whether lines and in the diagram below are parallel.
Using the converse of the Consecutive Interior Angles Theorem, you should be able to identify that if the two angles in the figure are supplementary, then lines and are parallel. We add the two consecutive interior angles to find their sum.
The two angles in the figure sum to so lines and are in fact parallel.
Parallel Lines Property
The last theorem to explore in this lesson is called the Parallel Lines Property. It is a transitive property. Does the phrase transitive property sound familiar? You have probably studied other transitive properties before, but usually talking about numbers. Examine the statement below.
If and , then
Notice that we used a property similar to the transitive property in a proof above. The Parallel Lines Property says that if line is parallel to line , and line is parallel to line , then lines and are also parallel. Use this information to solve the final practice problem in this lesson.
Example 6
Are lines and in the diagram below parallel?
Look at this diagram carefully to establish the relationship between lines and and lines and . Starting with line , the angle shown measures This angle is an alternate exterior angle to the angle labeled on line . Since the alternate exterior angles are congruent, these two lines are parallel. Next look at the relationship between and . The angle shown on line measures and it corresponds to the angle marked on line . Since the corresponding angles on these two lines are congruent, lines and are also parallel.
Using the Parallel Lines Property, we can identify that lines and are parallel, because is parallel to and is also parallel to .
Note that there are many other ways to reason through this problem. Can you think of one or two alternative ways to show ?
Lesson Summary
In this lesson, we explored how to work with the converse of theorems we already k
new. Specifically, we have learned:
How to identify and use the Corresponding Angles Converse Postulate.
How to identify and use the Converse of Alternate Interior Angles Theorem.
How to identify and use the Converse of Alternate Exterior Angles Theorem.
How to identify and use the Converse of Consecutive Interior Angles Theorem.
How to identify and use the Parallel Lines Property.
These will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply theorems and postulates to mathematical situations.
Points To Consider
You have now studied the many rules about parallel lines and the angles they form. In the next lesson, you will delve deeper into concepts of lines in the plane. You will apply some of the geometric properties of lines to slopes and graphing in the coordinate plane.
Review Questions
Solve each problem.
Are lines and parallel in the diagram below? If yes, how do you know?
Are lines and parallel in the following diagram? Why or why not?
Are lines and parallel in the diagram below? Why or why not?