The document provides an example of proving that two angles are congruent (<1 ≡ <2) given that two lines (P and Q) are parallel and cut by a transversal (l). It shows the step-by-step work, stating each statement and providing the corresponding postulate or theorem as the reason. It then provides a quiz with two examples to practice applying postulates and theorems to statements about angles and parallel lines.

1. PROVING PARALLELISM

2. GIVEN: P║Q TRANSVERSAL ℓ CUTS
P AND Q
Prove: <1 ≡ <2
Write the corresponding postulates as reasons:
Statements Reasons
1. p║q
2. <3 ≡ <2
3. <1 ≡ <3
4. <1 ≡ <2
1
3
2

3. GIVEN: P║Q TRANSVERSAL ℓ CUTS
P AND Q
Prove: <1 ≡ <2
Write the corresponding postulates as reasons:
Statements Reasons
1. p║q Given
2. <3 ≡ <2
3. <1 ≡ <3
4. <1 ≡ <2
1
3
2

4. GIVEN: P║Q TRANSVERSAL ℓ CUTS
P AND Q
Prove: <1 ≡ <2
Write the corresponding postulates as reasons:
Statements Reasons
1. p║q Given
2. <3 ≡ <2 Corresponding Angle Postulate
3. <1 ≡ <3
4. <1 ≡ <2
1
3
2

5. GIVEN: P║Q TRANSVERSAL ℓ CUTS
P AND Q
Prove: <1 ≡ <2
Write the corresponding postulates as reasons:
Statements Reasons
1. p║q Given
2. <3 ≡ <2 Corresponding Angle Postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <1 ≡ <2
1
3
2

6. GIVEN: P║Q TRANSVERSAL ℓ CUTS
P AND Q
Prove: <1 ≡ <2
Write the corresponding postulates as reasons:
Statements Reasons
1. p║q Given
2. <3 ≡ <2 Corresponding Angle Postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <1 ≡ <2 Transitive Property
1
3
2

7. GIVEN: P║Q TRANSVERSAL ℓ CUTS
P AND Q
Prove: <1 and <2 are supplementary
Write the corresponding postulates as reasons:
Statements Reasons
1. p║q Given
2. <2 and <3 are supplementary Linear Pair
3. <1 ≡ <3 Alternate Interior Angle
4. <1 and <2 are supplementary Linear Pair
1
32

8. LET’S TRY
Letter A, page 307. In the given figure, name the postulate or
theorem that justifies each statement, given m ║ n.

9. Statements Reasons
1. <2 ≡ <8
2. <3 ≡ <7
3. <1 ≡ <3
4. <3 and <8 are supplementary
5. <1 ≡ <7
6. <4 and <7 are supplementary
7. There exists a line through P
parallel to line n
8. a ┴b
1
32

10. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7
3. <1 ≡ <3
4. <3 and <8 are supplementary
5. <1 ≡ <7
6. <4 and <7 are supplementary
7. There exists a line through P
parallel to line n
8. a ┴b
1
32

11. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3
4. <3 and <8 are supplementary
5. <1 ≡ <7
6. <4 and <7 are supplementary
7. There exists a line through P
parallel to line n
8. a ┴b
1
32

12. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <3 and <8 are supplementary
5. <1 ≡ <7
6. <4 and <7 are supplementary
7. There exists a line through P
parallel to line n
8. a ┴b
1
32

13. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <3 and <8 are supplementary Same Side of the transversal are
supplementary
5. <1 ≡ <7
6. <4 and <7 are supplementary
7. There exists a line through P parallel to
line n
8. a ┴b
1
32

14. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <3 and <8 are supplementary Same Side of the transversal are
supplementary
5. <1 ≡ <7 Alternate Exterior Angles
6. <4 and <7 are supplementary
7. There exists a line through P parallel to
line n
8. a ┴b
1
32

15. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <3 and <8 are supplementary Same Side of the transversal are
supplementary
5. <1 ≡ <7 Alternate Exterior Angles
6. <4 and <7 are supplementary Same Side of the transversal are
supplementary
7. There exists a line through P parallel to
line n
8. a ┴b
1
32

16. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <3 and <8 are supplementary Same Side of the transversal are
supplementary
5. <1 ≡ <7 Alternate Exterior Angles
6. <4 and <7 are supplementary Same Side of the transversal are
supplementary
7. There exists a line through P parallel to
line n
Parallel Postulate
8. a ┴b
1
32

17. Statements Reasons
1. <2 ≡ <8 Alternate interior angles
2. <3 ≡ <7 Corresponding angle postulate
3. <1 ≡ <3 Vertical Angles Postulate
4. <3 and <8 are supplementary Same Side of the transversal are
supplementary
5. <1 ≡ <7 Alternate Exterior Angles
6. <4 and <7 are supplementary Same Side of the transversal are
supplementary
7. There exists a line through P parallel to
line n
Parallel Postulate
8. a ┴b Perpendicular Transversal Theorem
1
32

18. QUIZ Page 296-312

19. QUIZ:
1. Given: r║p prove <2 ≡ <3
Statements Reasons
1. r ║ s
2. <2 ≡ <1
3. <1 ≡ <3
4. <2 ≡ <3
1
2
3

20. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2
2. <3 ≡ <4
3. <1 and <5 are supplementary
4. <8 ≡ <9
5. <9 and <10 are supplementary
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

21. ANSWERS TO QUIZ Page 296-312

22. QUIZ:
1. Given: r║p prove <2 ≡ <3
Statements Reasons
1. r ║ s
2. <2 ≡ <1
3. <1 ≡ <3
4. <2 ≡ <3
1
2
3

23. QUIZ:
1. Given: r║p prove <2 ≡ <3
Statements Reasons
1. r ║ s Given
2. <2 ≡ <1
3. <1 ≡ <3
4. <2 ≡ <3
1
2
3

24. QUIZ:
1. Given: r║p prove <2 ≡ <3
Statements Reasons
1. r ║ s Given
2. <2 ≡ <1 Vertical Angles
3. <1 ≡ <3
4. <2 ≡ <3
1
2
3

25. QUIZ:
1. Given: r║p prove <2 ≡ <3
Statements Reasons
1. r ║ s Given
2. <2 ≡ <1 Vertical Angles
3. <1 ≡ <3 Same Side of the Transversal
4. <2 ≡ <3
1
2
3

26. QUIZ:
1. Given: r║p prove <2 ≡ <3
Statements Reasons
1. r ║ s Given
2. <2 ≡ <1 Vertical Angles
3. <1 ≡ <3 Same Side of the Transversal
4. <2 ≡ <3 Transitive Property
1
2
3

27. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2
2. <3 ≡ <4
3. <1 and <5 are supplementary
4. <8 ≡ <9
5. <9 and <10 are supplementary
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

28. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4
3. <1 and <5 are supplementary
4. <8 ≡ <9
5. <9 and <10 are supplementary
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

29. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary
4. <8 ≡ <9
5. <9 and <10 are supplementary
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

30. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9
5. <9 and <10 are supplementary
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

31. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9 Alternate Exterior Angles
5. <9 and <10 are supplementary
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

32. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9 Alternate Exterior Angles
5. <9 and <10 are supplementary Same Side of the Transversal (exterior)
6. <2 ≡ <3
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

33. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9 Alternate Exterior Angles
5. <9 and <10 are supplementary Same Side of the Transversal (exterior)
6. <2 ≡ <3 Linear Pairs
7. <5 and <7 are supplementary
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

34. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9 Alternate Exterior Angles
5. <9 and <10 are supplementary Same Side of the Transversal (exterior)
6. <2 ≡ <3 Linear Pairs
7. <5 and <7 are supplementary Linear Pairs
8. m<3 + m<9 = 180º
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

35. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9 Alternate Exterior Angles
5. <9 and <10 are supplementary Same Side of the Transversal (exterior)
6. <2 ≡ <3 Linear Pairs
7. <5 and <7 are supplementary Linear Pairs
8. m<3 + m<9 = 180º Definition of Supplementary Angles
9. <6 ≡ <9
3
18
2 9
54 7
1
0
6

36. QUIZ:
2. Refer to the figure. Name the postulate or theorem used in the
following statementStatements Reasons
1. <1 ≡ <2 Alternate interior Angles
2. <3 ≡ <4 Corresponding Angle
3. <1 and <5 are supplementary Same side of the transversal (interior)
4. <8 ≡ <9 Alternate Exterior Angles
5. <9 and <10 are supplementary Same Side of the Transversal (exterior)
6. <2 ≡ <3 Linear Pairs
7. <5 and <7 are supplementary Linear Pairs
8. m<3 + m<9 = 180º Definition of Supplementary Angles
9. <6 ≡ <9 Alternate Exterior Angles
3
18
2 9
54 7
1
0
6