Division of Segments and Angles: Midpoints, Bisectors, and Trisectors

1. Lesson 1.5 Division of Segments and Angles Objective: Identify midpoints and bisectors of segments, trisection points and trisectors of segments, angle bisectors and trisectors.

2. Definitions Def. A point (or segment, ray, or line) that divides a segment into two congruent segments bisects the segment. The bisection point is called the midpoint of the segment. Y A M B Note: Only segments have midpoints! X Why can’t a ray or line have a midpoint? X X is not a midpoint Y Y is not a midpoint

3. Example If D is the midpoint of segment FE, what conclusions can we draw? G F D E Conclusions:

4. Definitions A segment divided into three congruent parts is said to be trisected. Def. Two points (or segments, rays, or lines) that divides a segment into 3 congruent segments trisect the segment. The 2 points at which the segment is divided are called trisections points. Note: One again, only segments have trisection points!

5. Examples If , what conclusions can we draw? C S R A If E and F are trisection points of segment DG, what conclusions can we draw? H D G F E

6. Definitions Like a segment, angles can also be bisected and trisected. Def. A ray that divides an angle into 2 congruent angles bisects the angle. The dividing ray is called the angle bisector. Def. Two rays that divide an angle into 3 congruent angles trisects the angle. The 2 dividing rays are called angle trisectors.

7. Examples A D 40° 40° B C C A D 35° 35° 35° B E

8. Example #1 Does M bisect segment OP? 2x - 6 x + 8 O P M 44

9. Example #2 A Given: B is a midpoint of Prove: D B C Statement Reason

10. Example #3 Segment EH is divided by F and G in the ratio 5:3:2 from left to right. If EH = 30, find FG and name the midpoint of segment EH. G H F E

11. Classwork 1.1-1.3 Review Worksheet

12. Homework Lesson 1.5 Worksheet