1. Transversal and Parallel Lines
Angles formed by
Transversal and Parallel Lines
March 9, 2011

2. Warm Up 3/12/12

3. Warm Up 3/9/12

4. Parallel Lines are… …..coplanar lines that do not intersect.m
m || n n
Skew lines are non-coplanar, non-intersecting lines.
The only difference between skew and parallel is the coplanar part. What are the other ways to define parallel? p q

5. The Transversal Any line that intersects two or more coplanar lines. t

6. The Transversal t r
When lines intersect, angles are formed in several locations. s

7. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
Determine the two sets of angles that are congruent.
1, 4, 5, 8
2, 3, 6, 7
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

8. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
Angle pair relationships are formed.
Some angle pairs are congruent and other angle pairs are supplementary.
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

9. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
Congruent angles have the same measure.
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

10. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
Supplementary Angles are angles that have a sum of 180 degrees.
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

11. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
Certain angles are given "names" that describe "where" the angles are located in relation to the lines.
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

12. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
INTERIOR
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

13. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
EXTERIOR
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.
EXTERIOR

14. Corresponding Angles r s t 1 5
Corresponding angles are congruent angles on the same side of the transversal.
Postulate 10 cannot be proven, because it is the first angle relationship and so is assumed to be true. The other angle relationships can be proven, because of the assumption that corresponding angles are congruent. Either of the other two angle pairs could have been selected to serve as the assumption, and the others proven from that…the key here is that the choice was arbitrary.

15. Corresponding Angles . t r 2 6 s
Postulate 10 cannot be proven, because it is the first angle relationship and so is assumed to be true. The other angle relationships can be proven, because of the assumption that corresponding angles are congruent. Either of the other two angle pairs could have been selected to serve as the assumption, and the others proven from that…the key here is that the choice was arbitrary.

16. Corresponding Angles t r 3 s 7
Postulate 10 cannot be proven, because it is the first angle relationship and so is assumed to be true. The other angle relationships can be proven, because of the assumption that corresponding angles are congruent. Either of the other two angle pairs could have been selected to serve as the assumption, and the others proven from that…the key here is that the choice was arbitrary.

17. Corresponding Angles t r 4 s 8
Postulate 10 cannot be proven, because it is the first angle relationship and so is assumed to be true. The other angle relationships can be proven, because of the assumption that corresponding angles are congruent. Either of the other two angle pairs could have been selected to serve as the assumption, and the others proven from that…the key here is that the choice was arbitrary.

18. Alternate Interior Angles3 6
Alternate Interior angles are congruent angles on opposite sides of the transversal and inside the parallel lines.
Prove theorem 3-2 with them.

19. Alternate Interior Angles4 5
Prove theorem 3-2 with them.

20. Alternate Exterior Angles2 7
Alternate Exterior angles are congruent angles on opposite sides of the transversal and outside the parallel lines.
Prove theorem 3-2 with them.

21. Alternate Exterior Angles1 8
Prove theorem 3-2 with them.

22. Same Side Interior r s t 4 6
Same side interior angles are supplementary angles on the same side of the transversal and inside the parallel lines.
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

23. Same Side Interior r s t 3 5
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

24. Same Side Exterior r s t 2 8
Same side exterior angles are supplementary angles on the same side of the transversal and outside the parallel lines.
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

25. Same Side Exterior r s t 1 7
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

26. Vertical Angles r s t 1 2 3 4 5 6 7 8
Vertical angles are congruent angles located diagonally opposite each other.
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

27. Vertical Angles r s t 2 3
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

28. Vertical Angles r s t 5 8
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

29. Vertical Angles r s t 6 7
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

30. Angle 2 measures 110°. What other angles have the same measure?1. 2. 3. 4. 5. 6. 7. 8.

31. Answer: 1. 2. 3. 4. 5. 6. 7. 8.

32. What is the measure of ? 1. 2. 3. 4. 5. 6. 7. 8.

33. What is the measure of ?
Answer: 1. 2. 3. 4. 5. 6. 7. 8.

34. 9) Lines l and m are parallel. l||m Find the missing angles.
42° 2 3 4 l 5 6 7 8 m

35. 9) Lines l and m are parallel. l||m Find the missing angles.
42°
138°
138°
42° l
42°
138°
138°
42° m

36. 10) Lines l and m are parallel. l||m Find the missing angles.
81° 2 3 4 l 5 6 7 8 m

37. 10) Lines l and m are parallel. l||m Find the missing angles.
81°
99°
99°
81° l
81°
99°
99°
81° m

38. In the diagram below, j ║ k. What is m 1?
120°

39. Solution
m  ° = 180°
m 1 = 60°

40. Find the value for x
125° 4
(x + 15)°

41. Solution
m4 = 125°
m4 +(x+15)°=180°
125°+(x+15)°= 180°
x = 40