The document discusses how to prove lines are parallel using angles formed by a transversal. It provides four theorems involving corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles that can be used to show lines are parallel if the angles are congruent or supplementary. An example problem demonstrates finding values of x and y that make two sets of lines parallel by using the consecutive interior angles theorem. The document also includes a multi-step proof involving bisecting an angle and using the alternate interior angles theorem.

1. Proving Lines Parallel
The student is able to (I can):
• Use angles formed by a transversal to show that lines are
parallel.

2. Recall that the converse of a theorem is found by exchanging
(flipping) the hypothesis and conclusion.
The converses of the parallel line theorems can be used to
prove lines parallel.
• Corresponding Angles
• Alternate Interior Angles
• Alternate Exterior Angles
• Consecutive Interior Angles
– If angles are supplementary, then the lines are parallel.
If the angles are congruent,
then the lines are parallel.

3. Example Find values of x and y that make the red
lines parallel and the blue lines parallel.
If the blue lines are parallel, then the con-
secutive interior angles must be supple-
mentary.
x − 40 + x + 40 = 180
2x = 180
x = 90
(x−40)° (x+40)°
y°

4. Example Find values of x and y that make the red
lines parallel and the blue lines parallel.
If the red lines are parallel, then the con-
secutive interior angles must be supple-
mentary.
90 − 40 + y = 180
50 + y = 180
y = 130
(x−40)° (x+40)°
y°

5. Given: bisects ∠DBA; ∠3 ≅ ∠1
Prove:
Plan of proof: Because bisects ∠DBA, ∠2 ≅ ∠3. Because
∠3 ≅ ∠1, I can set ∠2 ≅∠1 using substitution.
∠1 and ∠2 are alternate interior angles, and
since they are congruent, the lines are parallel.
BE
CD BE
C
32
1
ED A
B
BE

6. Given: bisects ∠DBA; ∠3 ≅ ∠1
Prove:
BE
CD BE
C
32
1
ED A
B
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. bisects ∠DBA 1. GivenBE

7. Given: bisects ∠DBA; ∠3 ≅ ∠1
Prove:
BE
CD BE
C
32
1
ED A
B
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. bisects ∠DBA 1. Given
2. ∠2 ≅ ∠3 2. Def. ∠ bisector
BE

8. Given: bisects ∠DBA; ∠3 ≅ ∠1
Prove:
BE
CD BE
C
32
1
ED A
B
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. bisects ∠DBA 1. Given
2. ∠2 ≅ ∠3 2. Def. ∠ bisector
3. ∠3 ≅ ∠1 3. Given
BE

9. Given: bisects ∠DBA; ∠3 ≅ ∠1
Prove:
BE
CD BE
C
32
1
ED A
B
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. bisects ∠DBA 1. Given
2. ∠2 ≅ ∠3 2. Def. ∠ bisector
3. ∠3 ≅ ∠1 3. Given
4. ∠2 ≅ ∠1 4. Substitution prop. ≅
BE

10. Given: bisects ∠DBA; ∠3 ≅ ∠1
Prove:
BE
CD BE
C
32
1
ED A
B
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. bisects ∠DBA 1. Given
2. ∠2 ≅ ∠3 2. Def. ∠ bisector
3. ∠3 ≅ ∠1 3. Given
4. ∠2 ≅ ∠1 4. Substitution prop. ≅
5. 5. Converse of alt. int. ∠s
BE
CD BE