1.5 Division of Segments and Angles

1. 1.5 Division of Segments and Angles

2. Bisector • A point (or segment, ray, or line) that divides a segment into two congruent segments bisects the segment. • This point is called the midpoint • Only segments have midpoints

3. Examples Y A B C • If XY bisects AC at B, what conclusions can we draw? 2. If D is the midpoint of FE, what conclusions can we draw? 3. If OK=KP, what conclusions can we draw? X G F D E O K M J P

5. Trisection Points and Trisecting a Segment • Two points (or segments, rays, or lines) that divide a segment into three congruent segments trisect the segment. The two points at which the segment is divided are called the trisection points of the segment. • Only segments!

6. Examples • If AR=RS=SC, what conclusions can we draw? • If E and F are trisection points of DG, what conclusions can we draw?

8. Angle Bisectors • A ray that divides an angle into two congruent angles bisects the angle. The dividing ray is called the bisector of the angle. Angle Trisectors • Two rays that divide an angle into three congruent angles trisects the angle. The two dividing rays are called the trisectors of the angle.

10. Statements We Can Use To Prove page If a ray bisects an angle, it divides the angle into two congruent angles (Def. of Angle Bisector) If a line/point bisects a segment, it divides the segment into two congruent segments (Def. of Bisector) If a point divides a segment into two congruent segments, it is the midpoint of the segment. (Def. of Midpoint) If two rays divide an angle into three congruent angle, they are trisectors of the angle (Def. of Angle Trisector) If two rays trisect an angle, they divide the angle into three congruent parts (Def. of Angle Trisector)

11. Proofs! Given: PS bisects <RPO Prove: <RPS=<OPS

12. And More! Given: DH=HF Prove: H is the midpoint of DF

13. Example Problems (not proofs) Done in class!