This document discusses ways to prove that two lines are parallel using angle relationships. It introduces the converse of corresponding angles postulate, parallel postulate, and converses of alternate exterior angles, consecutive interior angles, and alternate interior angles postulates. Examples are provided to demonstrate using these theorems to prove lines are parallel or not parallel based on given angle information. Readers are asked to complete practice problems applying these concepts.
1. Section 3-5
Proving Lines Parallel
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2. Essential Questions
• How do you recognize angle pairs tha
occur with parallel lines?
• How do you prove that two lines are
parallel using angle relationships?
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3. Postulates & Theorems
1. Converse of Corresponding Angles Postulate
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4. Postulates & Theorems
1. Converse of Corresponding Angles Postulate
If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines are
parallel.
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6. Postulates & Theorems
2. Parallel Postulate
If given a line and a point not on the line, then there
exists exactly one line through the point that is parallel
to the given line.
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8. Postulates & Theorems
3. Alternate Exterior Angles Converse
If two lines in a plane are cut by a transversal so that a
pair of alternate exterior angles is congruent, then the
two lines are parallel.
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10. Postulates & Theorems
4. Consecutive Interior Angles Converse
If two lines in a plane are cut by a transversal so that a
pair of consecutive interior angles is supplementary,
then the lines are parallel.
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12. Postulates & Theorems
5. Alternate Interior Angles Converse
If two lines in a plane are cut by a transversal so that a
pair of alternate interior angles is congruent, then the
two lines are parallel.
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14. Postulates & Theorems
6. Perpendicular Transversal Converse
In a plane, if two lines are perpendicular to the same
line, then they are parallel.
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15. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
a. ∠1≅ ∠3
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16. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
a. ∠1≅ ∠3
a b since these
congruent angles are
also corresponding,
the Corresponding
Angles Converse
holds
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17. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
b. m∠1= 103° and m∠4 = 100°
∠1 and ∠4
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18. Example 1
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
b. m∠1= 103° and m∠4 = 100°
∠1 and ∠4 are
alternate interior
angles, but not
congruent, so a is not
parallel to c
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19. Example 2
Find m∠ZYN so that PQ MN.
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20. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
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21. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
4x = 60
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22. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
4x = 60
x = 15
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23. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
4x = 60
x = 15
m∠ZYN = 7(15) + 35
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24. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
4x = 60
x = 15
m∠ZYN = 7(15) + 35 = 105 + 35
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25. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
4x = 60
x = 15
m∠ZYN = 7(15) + 35 = 105 + 35 = 140
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26. Example 2
Find m∠ZYN so that PQ MN.
11x − 25 = 7x + 35
4x = 60
x = 15
m∠ZYN = 7(15) + 35 = 105 + 35 = 140
m∠ZYN = 140°
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