Geometric Proofs:
Proving Lines are Parallel
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You have already learned how to that certain angle pairs are congruent if you have parallel lines. Now you will have to
use the same logic to work backwards to prove that lines are parallel. Use the following diagram for each proof.
Part 1 – “Mini Proofs” – Proving Lines are Parallel in 2 Steps
1. Given:
1
8
Prove: m || n
2. Given:
1
5
Prove: m || n
3. Given:
2
6
Prove: m || n
4. Given:
2
7
Prove: m || n
Statements
Reasons
Statements
Reasons
Statements
Reasons
Statements
Reasons
m
n
1
2
3
4
5
6
7
8
m
n
Copyright 2013 – Dana Burrows
5. Given:
3
6
Prove: m || n
6. Given:
3
7
Prove: m || n
7. Given:
4
5
Prove: m || n
8. Given:
4
8
Prove: m || n
9. Given:
4 and
6 are
supplementary
Prove: m || n
10. Given:
3 and
5 are
supplementary
Prove: m || n
Statements
Reasons
Statements
Reasons
Statements
Reasons
Statements
Reasons
Statements
Reasons
Statements
Reasons
Copyright 2013 – Dana Burrows
Part 2 – Proving Lines Parallel in Multiple Steps
11. Given: m
3 = m
7
Prove: m || n
12. Given: m
3 + m
5 = 180 degrees
Prove: m || n
13. Given: m
1 = 29 degrees
m
8 = 29 degrees
Prove: m || n
14. Given: m
4 = 103 degrees
m
5 = 103 degrees
Prove: m || n
Statements
Reasons
Statements
Reasons
Statements
Reasons
Statements
Reasons
Copyright 2013 – Dana Burrows
Part 1 – “Mini Proofs” – Proving Lines are Parallel in 2 Steps
1. Given:
1
8
Prove: m || n
2. Given:
1
5
Prove: m || n
3. Given:
2
6
Prove: m || n
4. Given:
2
7
Prove: m || n
5. Given:
3
6
Prove: m || n
Statements
Reasons
1
8
Given
m || n
Converse of the Alternate Exterior Angles Theorem
Statements
Reasons
1
5
Given
m || n
Converse of the Corresponding Angles Postulate
Statements
Reasons
2
6
Given
m || n
Converse of the Corresponding Angles Postulate
Statements
Reasons
2
7
Given
m || n
Converse of Alternate Exterior Angles Theorem
Statements
Reasons
3
6
Given
m || n
Converse of the Alternate Interior Angles Theorem
Copyright 2013 – Dana Burrows
6. Given:
3
7
Prove: m || n
7. Given:
4
5
Prove: m || n
8. Given:
4
8
Prove: m || n
9. Given:
4 and
6 are
supplementary
Prove: m || n
10. Given:
3 and
5 are
supplementary
Prove: m || n
Statements
Reasons
3
7
Given
m || n
Converse of the Corresponding Angles Postulate
Statements
Reasons
4
5
Given
m || n
Converse of the Alternate Interior Angles Theorem
Statements
Reasons
4
8
Given
m || n
Converse of the Corresponding Angles Postulate
Statements
Reasons
4 and
6 are
supplementary
Given
m || n
Converse of the Same-Side Interior Angles Postulate
(a.k.a. the Converse of the Consecutive Interior
Angles Theorem)
Statements
Reasons
3 and
5 are
supplementary
Given
m || n
Converse of the Same-Side Interior Angles Postulate
(a.k.a. the Converse of the Consecutive Interior
Angles Theorem)
Copyright 2013 – Dana Burrows
Part 2 – Proving Lines Parallel in Multiple Steps
11. Given: m
3 = m
7
Prove: m || n
12. Given: m
3 + m
5 = 180 degrees
Prove: m || n
13. Given: m
1 = 29 degrees
m
8 = 29 degrees
Prove: m || n
14. Given: m
4 = 103 degrees
m
5 = 103 degrees
Prove: m || n
Statements
Reasons
m
3 = m
7
Given
3
7
Definition of Congruent Angles
m || n
Converse of the Corresponding Angles Postulate
Statements
Reasons
m
3 + m
5 = 180
degrees
Given
3 and
5 are
supplementary
Definition of Supplementary Angles
m || n
Converse of the Corresponding Angles Postulate
Statements
Reasons
m
1 = 29 degrees
Given
m
7 = 29 degrees
Given
m
1 = m
7
Transitive or Substitution
1
7
Definition of Congruent Angles
m || n
Converse of the Alternate Exterior Angles Theorem
Statements
Reasons
m
4 = 103 degrees
Given
m
5 = 103 degrees
Given
m
4 = m
5
Transitive or Substitution
4
5
Definition of Congruent Angles
m || n
Converse of the Alternate Interior Angles Theorem
