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Alternate Interior Angles
Bill Zahner
Lori Jordan
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Printed: May 4, 2016
AUTHORS
Bill Zahner
Lori Jordan
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C HAPTER
Chapter 1. Alternate Interior Angles
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Alternate Interior Angles
Here you’ll learn what alternate interior angles are and their relationship with parallel lines.
What if you were presented with two angles that are on opposite sides of a transversal, but inside the lines? How
would you describe these angles and what could you conclude about their measures? After completing this Concept,
you’ll be able to answer these questions using your knowledge of alternate interior angles.
Watch This
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CK-12 Foundation: Chapter3AlternateInteriorAnglesA
Watch the portions of this video dealing with alternate interior angles.
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URL: https://www.ck12.org/flx/render/embeddedobject/1328
James Sousa: Angles and Transversals
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URL: https://www.ck12.org/flx/render/embeddedobject/1339
James Sousa: Proof that Alternate Interior Angles Are Congruent
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James Sousa: Proof of Alternate Interior Angles Converse
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Guidance
Alternate Interior Angles are two angles that are on the interior of l and m, but on opposite sides of the transversal.
6 3 and 6 6 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: l || m
Prove: 6 3 ∼
=6 6
TABLE 1.1:
Statement
1. l || m
2. 6 3 ∼
=6 7
6
3. 7 ∼
=6 6
4. 6 3 ∼
=6 6
Reason
Given
Corresponding Angles Postulate
Vertical Angles Theorem
Transitive PoC
There are several ways we could have done this proof. For example, Step 2 could have been 6 2 ∼
= 6 6 for the same
∼
∼
6
6
6
6
reason, followed by 2 = 3. We could have also proved that 4 = 5.
Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and alternate interior angles
are congruent, then the lines are parallel.
Example A
Find m6 1.
2
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Chapter 1. Alternate Interior Angles
m6 2 = 115◦ because they are corresponding angles and the lines are parallel.
m6 1 = 115◦ also.
6
6
1 and 6 2 are vertical angles, so
1 and the 115◦ angle are alternate interior angles.
Example B
Find the measure of the angle and x.
The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and
solve for x.
(4x − 10)◦ = 58◦
4x = 68◦
x = 17◦
Example C
Prove the Converse of the Alternate Interior Angles Theorem.
Given: l and m and transversal t
6 3∼
=6 6
Prove: l || m
TABLE 1.2:
Statement
1. l and m and transversal t 6 3 ∼
=6 6
Reason
Given
3
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TABLE 1.2: (continued)
Statement
2. 6 3 ∼
=6 2
6
3. 2 ∼
=6 6
4. l || m
Reason
Vertical Angles Theorem
Transitive PoC
Converse of the Corresponding Angles Postulate
Watch this video for help with the Examples above.
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CK-12 Foundation: Chapter3AlternateInteriorAnglesB
Guided Practice
1. Is l || m?
2. What does x have to be to make a || b?
3. List the pairs of alternate interior angles:
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Chapter 1. Alternate Interior Angles
Answers:
1. First, find m6 1. We know its linear pair is 109◦ . By the Linear Pair Postulate, these two angles add up to 180◦ , so
m6 1 = 180◦ − 109◦ = 71◦ . This means that l || m, by the Converse of the Corresponding Angles Postulate.
2. Because these are alternate interior angles, they must be equal for a || b. Set the expressions equal to each other
and solve.
3x + 16◦ = 5x − 54◦
70◦ = 2x
35◦ = x
To make a || b, x = 35◦ .
3. Alternate Interior Angles: 6 4 and 6 5, 6 3 and 6 6.
Interactive Practice
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Explore More
1. Is the angle pair 6 6 and 6 3 congruent, supplementary or neither?
2. Give two examples of alternate interior angles in the diagram:
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For 3-4, find the values of x.
3.
4.
For question 5, use the picture below. Find the value of x.
5. m6 4 = (5x − 33)◦ , m6 5 = (2x + 60)◦
6. Are lines l and m parallel? If yes, how do you know?
For 7-12, what does the value of x have to be to make the lines parallel?
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7.
8.
9.
10.
11.
12.
m6
m6
m6
m6
m6
m6
Chapter 1. Alternate Interior Angles
4 = (3x − 7)◦ and m6 5 = (5x − 21)◦
3 = (2x − 1)◦ and m6 6 = (4x − 11)◦
3 = (5x − 2)◦ and m6 6 = (3x)◦
4 = (x − 7)◦ and m6 5 = (5x − 31)◦
3 = (8x − 12)◦ and m6 6 = (7x)◦
4 = (4x − 17)◦ and m6 5 = (5x − 29)◦
For questions 13-15, use the picture below.
13. What is the alternate interior angle to 6 4?
14. What is the alternate interior angle to 6 5?
15. Are the two lines parallel? Explain.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 3.4.
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