1. 3.1: Lines and Angles 1

2. Today’s Objectives  To identify relationships between figures in space  To identify angles formed by two lines and a transversal

3. Parallel Lines  Parallel lines are coplanar lines that do not intersect.  Arrows are used to indicate lines are parallel.  The symbol used for parallel lines is ||. 3

4. Skew Lines and Parallel Planes  Two lines are skew if they do not intersect and are not in the same plane (not coplanar). Ex: are skew.  All planes are either parallel or intersecting.  Parallel planes are two planes that do not intersect. 4 Ex: Plane ABC and Plane EFG are parallel planes

5. Examples: 1.Name all segments that are parallel to 2.Name all segments that intersect 3.Name all segments that are skew to 4.Name all planes that are parallel to plane BCG. Answers: 1.Segments BC, FG, & EH. 2.Segments DH, DC, AE & AB. 3.Segments CG, BF, FE, & GH. 4.Plane ADH. 5

6. Example: 6 1) What are the lines parallel to CD? 2) What lines are skew to CD? 3) Name the parallel plane to CDA.

7. 7 Transversal Definition: A line that intersects two or more lines in a plane at different points is called a transversal. When a transversal t intersects line n and m, EIGHT angles are formed Exterior angles: Outside the lines Interior angles : Between the lines t m n

8. 8 Corresponding Angles Corresponding Angles: Two angles, on the same side of the transversal, that occupy corresponding positions, one interior and one exterior.  2 and  6,  1 and  5,  3 and  7,  4 and  8 12 34 56 78

9. 9 Alternate Angles Alternate Interior Angles: Two angles that lie between the lines on opposite sides of the transversal (but not a linear pair). Alternate Exterior Angles: Two angles that lie outside the lines on opposite sides of the transversal.  3 and  6,  4 and  5  2 and  7,  1 and  8 12 34 56 78

10. 10 Consecutive Angles (aka Same-Side) Consecutive Interior Angles: Two angles that lie between the lines, both on the same side of the transversal. Consecutive Exterior Angles: Two angles that lie outside the lines, both on the same side of the transversal.  3 and  5,  4 and  6  1 and  7,  2 and  8 12 34 56 78

11. 11 Vertical Angles & Linear Pair Vertical Angles: Linear Pair:  1   4,  2   3,  5   8,  6   7 Two angles that are opposite angles. Vertical angles are congruent.  1 &  2,  2 &  4,  4 &  3,  3 &  1,  5 &  6,  6 &  8,  8 &  7,  7 &  5 Supplementary angles that form a straight line (sum = 180  ) 12 34 56 78

12. Example List all pairs that fit the description a.Corresponding b.Alternate Exterior c.Alternate Interior d.Consecutive Interior < 4 and < 2 < 3 and < 1 < 5 and < 7 < 6 and < 8 < 2 and < 6 < 3 and < 7 < 4 and < 8 < 1 and < 5 < 3 and < 2 < 6 and < 7

13. Example Complete the statement with corresponding, alternate exterior, alternate interior, or consecutive interior. 1.< 4 and < 8 are 2.< 2 and < 6 are 3.< 1 and < 8 are 4.< 7 and < 2 are 5.< 4 and < 6 are 6.< 5 and < 7 are Alternate interior Consecutive exterior Corresponding Vertical Alternate exterior Consecutive interior

14. Take Home Message  Parallel and skew lines  When a transversal intersects two other lines, 8 different angles are formed  Alternate vs Consecutive (same-side)  Interior vs. Exterior