1. 3.1 PARALLEL LINES Thompson
How can I recognize planes, transversals, pairs of angles formed by a transversal, and parallel lines?

2. What You Should Learn Why You Should Learn It
Goal 1: How to identify angles formed by two lines and a transversal
Goal 2: How to use properties of parallel lines

3. Angles Formed by a Transversal
Transversal – a line that intersects two or more coplanar lines at different points
In the figure, the transversal t intersects the lines L and M t L M

4. Corresponding Angles t
Two angles are corresponding angles if they occupy corresponding positions, such as t 4 1 3 2 L M 5 8 6 7

5. Alternate Interior Angles
Two angles are alternate interior angles if they lie between L and M on opposite sides of t, such as t 4 1 3 2 L M 5 8 6 7

6. Alternate Exterior Angles
Two angles are alternate exterior angles if they lie outside L and M on opposite sides of t, such as t 4 1 3 2 L M 5 8 6 7

7. Same-Side Interior Angles (consecutive)
Two angles are same-side interior angles if they lie between L and M on the same side of t, such as t 4 1 3 2 L M 5 8 6 7

8. Example
Classify the pair of numbered angles.

9. These definitions depend on what line is the transversal.
Before you define the angle terms, you need to think about which line is the transversal!
Even within the same diagram, the transversal can change. This will often lead to “ignoring” other parts of the diagram (as we’ll see next…)

10. Example 1 Naming Pairs of Angles
How is related to the other angles? n 4 1 m 3 2 10 11 9 12 8 7 5 L 6

11. Example 1 Naming Pairs of Angles
How is related to the other angles?
are a linear pair. So are
are vertical angles
are alternate exterior angles. So are
are corresponding angles. So are 1 2 3 4 5 6 7 8 9 10 11 12 n m L

12. PARALLEL LINES AND CONGRUENT ANGLES
While these definitions apply whenever two lines are cut by a transversal, we will normally talk about this for parallel lines
1. What is the Parallel Postulate and how can I use it?
What are the pairs of congruent angles formed by parallel lines cut by a transversal?
Lets reexamine the angle definitions for parallel lines

13. Parallel Lines Two coplaner lines that do not intersect
The book notes them with “matching arrows” see page 117

14. TRANSVERSAL (Parallel)
A line that intersects two coplanar lines in two distinct points.

15. You see this occurrence everywhere!

16. SPECIAL ANGLES (parallel)2 1 4
Interior Angles – lie between the two lines (3, 4, 5, and 6) 3 6 5 8 7
Alternate Interior Angles – are on opposite sides of the transversal. (3 & 6 AND 4 and 5)
Same-Side Interior Angles – are on the same side of the transversal. (3 & 5 AND 4 & 6)

17. More Special Angles
Exterior Angles – lie outside the two lines (1, 2, 7, and 8) 2 1 4 3 6 5 8 7
Alternate Exterior Angles – are on opposite sides of the transversal (1& 8 AND 2 & 7)
Corresponding Angles – same location, different intersections (2 & 6, 4 & 8, 1 & 5, 3 & 7)

18. Parallel Lines and a Transversal
1. On your paper, construct two parallel lines, then construct an “angled” transversal.
Label each angle made (1 - 8)
Based on appearance, make a conjecture as to which angles are congruent, supplementary, or complementary to each other.
Using the protractor, measure and label each angle.
Make 4 conjectures about the angle pairs: ( alternate int, consecutive ext, corresponding…)

19. If 2 parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.

20. Vertical Angles & Linear Pair
Two angles that are opposite angles. Vertical angles are congruent.
 1   4,  2   3,  5   8,  6   7
Supplementary angles that form a line (sum = 180)
1 & 2 , 2 & 4 , 4 &3, 3 & 1,
5 & 6, 6 & 8, 8 & 7, 7 & 5 1 2 3 4 5 6 7 8

21. If two lines are Parallel:
1. A pair of corresponding s are .
2. A pair of alternate interior s are .
3. A pair of alternate exterior s are .
4. A pair of consecutive interior s are supplementary.
5. A pair of consecutive exterior s are supplementary.

22. Corresponding Angles & Consecutive Angles
Corresponding Angles: Two angles that occupy corresponding positions.
 2   6,  1   5,  3   7,  4   8 1 2 3 4 5 6 7 8

23. Consecutive Angles 1 2 3 4 5 6 7 8 m3 +m5 = 180º, m4 +m6 = 180º
Consecutive Interior Angles: Two angles that lie between parallel lines on the same sides of the transversal.
Consecutive Exterior Angles: Two angles that lie outside parallel lines on the same sides of the transversal.
m3 +m5 = 180º, m4 +m6 = 180º 1 2
m1 +m7 = 180º, m2 +m8 = 180º 3 4 5 6 7 8

24. Alternate Angles  3   6,  4   5  2   7,  1   8 1 2 3 4 5 6
Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair).
Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal.
 3   6,  4   5
 2   7,  1   8 1 2 3 4 5 6 7 8

25. Example: If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m< 1 = 100°. Justify your answers. t 16 15 14 13 12 11 10 9 8 7 6 5 3 4 2 1 s D C B A

26. Example
On the map below, 1st and 2nd Ave. are parallel.

27. Example
A city planner proposes to locate a small garden on the triangular island formed by the intersections of the four streets below.

28. Example
What are the measures of the three angles of the garden?

29. Example 5
Find the value of x.

30. Investigating Postulates
Construct a line named AB.
Somewhere above or below the line, put a pt P.
Construct 2 lines that go through pt P that are also || to AB

31. Through a point not on a line there is exactly ONE parallel to the given line.
Parallel Postulates

32. Investigating Postulates
Construct 2 parallel lines AB and CD.
Construct LP so that it is  to AB and also passes through CD.
Measure the angles of the intersection on LP and CD.
What can you conclude?

33. In a plane, if a line is perpendicular to 1 of 2 parallel lines, then it is perpendicular to the other.

34. Investigating Postulates
Construct 2 parallel lines AB and CD.
Construct FG so that it is || to CD
What can you conclude about AB and FG?
This is an example of what property?!

35. If 2 lines are parallel to a 3rd line, then they are parallel to each other. (transitive prop. of || lines)

36. Classwork - Go! Starting on Pg 118
5-8
11-17
19-26 30