1. Angles and Parallel Lines

2. Three ways in which two lines may be situated in space
They may fail to intersect and fail to be coplanar: L1 and L3 Called skew
They may intersect in a point: L1 and L2 In which case they are co-planar (Given two intersecting lines, there is exactly one plane containing both.)
They may be coplanar without intersecting each other: L2 and L3 Called parallel

3. Definitions: Skew and Parallel
Two lines are parallel if:
They are coplanar
They do not intersect
We write
On sketch

4. Definition: Transversal
A transversal of two lines is a line which intersects them in two different points.
Transversal
Not Transversal

5. Transversal
Transversals are often studied in relationship to parallel lines
When a transversal intersects two lines eight angles are formed: 1 2 3 4 5 6 7 8

6. Classification of angles
Corresponding Angles: Angles that occupy corresponding positions. 1 2 3 4 5 6 7 8

7. Classification of angles
Corresponding Angles: Angles that occupy corresponding positions.
Alternate: Angles that are on opposite sides of the transversal. 1 2 3 4 5 6 7 8

8. Classification of angles
Corresponding Angles: Angles that occupy corresponding positions.
Alternate: Angles that are on opposite sides of the transversal.
Same Side Angles: Angles that are on the same sides of the transversal. 1 2 3 4 5 6 7 8

9. Classification of angles
Corresponding Angles: Angles that occupy corresponding positions.
Alternate: Angles that are on opposite sides of the transversal.
Same Side Angles: Angles that are on the same sides of the transversal.
Interior: Angles that are between the parallel lines. 3 4 5 6

10. Classification of angles
Corresponding Angles: Angles that occupy corresponding positions.
Alternate: Angles that are on opposite sides of the transversal.
Same Side Angles: Angles that are on the same sides of the transversal.
Interior: Angles that are between the parallel lines. 1 2
Exterior: Angles that are outside the parallel lines. 7 8

11. F Corresponding Angles  2   6,  1   5,  3   7,  4   8
Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the Corresponding angles are congruent.
 2   6,  1   5,  3   7,  4   8 F 1 2 3 4 5 6 7 8

12. Alternate Angles
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the Alternate Interior Angles are congruent.
 3   6,  4   5 N Z 1 2 3 4 5 6 7 8

13. Alternate Angles
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the Alternate Exterior Angles are congruent.
 2   7,  1   8 1 2 3 4 5 6 7 8

14. Same Side or Consecutive Angles
Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then the Consecutive Interior Angles are supplementary.
m3 +m5 = 180º, m4 +m6 = 180º U 1 2 3 4 5 6 7 8

15. Lesson 2-4: Angles and Parallel Lines
Example
Lesson 2-4: Angles and Parallel Lines

16. Lesson 2-4: Angles and Parallel Lines
Homework
HW p 157 #10 – 16 even, 23, 27 – 30 all
Lesson 2-4: Angles and Parallel Lines