1. 3.1 Lines and Angles Mrs. Spitz Fall 2004

2. Standard/Objectives: Standard 3: Students will have a foundation in geometric concepts. Objectives: Identify relationships between lines. Identify angles formed by transversals.

3. Homework Assignment Pp. 132-133 #1-31

4. Definitions Parallel lines – Two lines are parallel lines if they are coplanar and do not intersect. Skew lines—Lines that do not intersect and are not coplanar. Parallel planes—two planes that do not intersect.

5. Example 1: Identifying relationships in space Think of each segment in the diagram. Which appear to fit the description? a. Parallel to AB and contains D b. Perpendicular to AB and contains D c. Skew to AB and contains D d. Name the plane(s) that contains D and appear to be parallel to plane ABE  A B D C F E H G

6. Solution a. CD, GH and EF are all parallel to AB, but only CD passes through D and is parallel to AB. b. BC, AD, AE and BF are all perpendicular to AB, but only AD passes through D and is perpendicular to AB c. DG, DH, and DE all pass through D and are skew to AB. d. Only plane DCH contains D and is parallel to plane ABE

7. Postulate 13: Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P l

8. Postulate 14: Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the given point perpendicular to the given line. l P

9. Perpendicular Construction—pg. 130 Use the following steps to construct a line that passes through a given point P and is perpendicular to a given line l. 1. Place the compass at point P and draw an arc that intersects the line l twice. Label the intersections A and B. 2. Draw an art with center A. Using the same radius, draw an arc with center B. Label the intersection of the arcs Q. 3. Use a straight edge to draw PQ. PQ  l.

10. Definitions: Transversal: a line that intersects two or more coplanar lines at different points. Angles 1 and 5 are corresponding angles Angles 1 and 8 are alternate exterior angles Angles 3 and 5 are alternate interior angles. 3 and 5 are consecutive interior angles 4 7 1 2 3 56 8

11. Example 2: Identifying Angle relationships List all pairs of angles that fit the description. a. Corresponding b. Alternate exterior c. Alternate interior d. Consecutive interior 1 2 3 4 7 6 5 8

12. Solution: a.  1 and  5;  2 and  6;  3 and  7,  4 and  8 b.  1 and  8,  2 and  7 c.  3 and  6,  4 and  5 d.  3 and  5,  4 and  6