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Parallel Lines and Transversals
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C ONCEPT
1
1
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.
Concept 1. Parallel Lines and Transversals
2
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• 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—6 6 and 6 8, 6 7 and 6 1, 6 5 and 6 3, and 6 4 and 6 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—6 7 and 6 3,
and 6 8 and 6 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—6 1 and 6 5, and 6 2
and 6 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—6 7 and 6 8 and 6 3 and 6 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 6 2 and 6 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 pand qare parallel. What is the measure of 6 1?
Because lines p and q are parallel, the 120◦ angle and 6 1 are corresponding angles, we know by the Corresponding
Angles Postulate that they are congruent. Therefore, m6 1 = 120◦ .
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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 6 1 is 120◦ , you can find the measurement of all the other angles. For example, 6 1
and 6 2 must be supplementary (sum to 180◦ ) because together they are a linear pair (we are using the Linear Pair
Postulate here). So, to find m6 2, subtract 120◦ from 180◦ .
m6 2 = 180◦ − 120◦
m6 2 = 60◦
So, m6 2 = 60◦ . Knowing that 6 2 and 6 3 are also supplementary means that m6 3 = 120◦ , since 120 + 60 = 180.
If m6 3 = 120◦ , then m6 4 must be 60◦ , because 6 3 and 6 4 are also supplementary. Notice that 6 1 ∼
= 6 3 (they both
◦
◦
∼
measure 120 ) and 6 2 = 6 4 (both measure 60 ). 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 land min the diagram below are parallel. What are the measures of angles αand β?
Concept 1. Parallel Lines and Transversals
4
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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◦ . Angle β corresponds to the 80◦ angle. So by the Corresponding
Angles Postulate, m6 β = 80◦ .
Now, because 6 α is made by the same intersecting lines and is opposite the 80◦ angle, these two angles are vertical
angles. Since you already learned that vertical angles are congruent, we conclude m6 α = 80◦ . Finally, compare
angles α and β. They both measure 80◦ , 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: AB and CD are parallel lines crossed by transversal 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 6 DW Z ∼
=
6 W ZA.
TABLE 1.1:
Statement
←
→ ←
→
1. ABkCD
2. 6 DW Z ∼
= 6 BZX
∼
3. 6 BZX = 6 W ZA
4. 6 DW Z ∼
= 6 W ZA
Reason
1.Given
2. Corresponding Angles Postulate
3. Vertical Angles Theorem
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 gand hin the diagram below are parallel. If m6 4 = 43◦ , what is the measure of 6 5?
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5
You know from the problem that m6 4 = 43◦ . That means that 6 40 s corresponding angle, which is 6 3, will measure
43◦ as well.
The corresponding angle you just filled in is also vertical to 6 5. Since vertical angles are congruent, you can conclude
m6 5 = 43◦ .
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 rand sin the diagram below are parallel. If the angle corresponding to 6 1measures 76◦ , what is m6 2?
Concept 1. Parallel Lines and Transversals
6
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This process should now seem familiar. The given 76◦ angle is adjacent to 6 2 and they form a linear pair. Therefore,
the angles are supplementary. So, to find m6 2, subtract 76◦ from 180◦ .
m6 2 = 180 − 76
m6 2 = 104◦
This example shows that if two parallel lines are cut by a transversal, the consecutive interior angles are supplementary; they sum to 180◦ . 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.
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Review Questions
Solve each problem.
←
→
←
→
Use the diagram below for Questions 1-4. In the diagram, lines AB and CD are parallel.
1. What term best describes the relationship between 6 AFG and 6 CGH?
a.
b.
c.
d.
alternate exterior angles
consecutive interior angles
corresponding angles
alternate interior angles
2. What term best describes the mathematical relationship between 6 BFG and 6 DGF?
a.
b.
c.
d.
congruent
supplementary
complementary
no relationship
3. What term best describes the relationship between 6 FGD and 6 AFG?
a.
b.
c.
d.
alternate exterior angles
consecutive interior angles
complementary
alternate interior angles
4. What term best describes the mathematical relationship between 6 AFE and 6 CGH?
a.
b.
c.
d.
congruent
supplementary
complementary
no relationship
Use the diagram below for questions 5-7. In the diagram, lines l and m are parallel γ, β, θ represent the measures of
the angles.
Concept 1. Parallel Lines and Transversals
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5. What is γ?
6. What is β?
7. What is θ?
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.
8.
9.
10.
11.
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 6 1 and 6 2 are supplementary.
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Review Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
c
a
d
b
73◦
107◦
107◦
65◦
65◦
115◦
Proof of Consecutive Interior Angle Theorem. Given r||s, prove 6 1 and 6 2 are supplementary.
TABLE 1.2:
Statement
1. r||s
2. 6 1 ∼
=6 3
6
3. 2 and 6 3 are supplementary
4. m6 2 + m6 3 = 180◦
5. m6 2 + m6 1 = 180◦
6. 6 2 and 6 1 are supplementary
Reason
1. Given
2. Corresponding Angles Postulate
3. Linear Pair Postulate
4. Definition of supplementary angles
5. Substitution (6 1 ∼
= 6 3)
6. Definition of supplementary angles
Concept 1. Parallel Lines and Transversals
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