1. GBK G EOMETRY Jordan Johnson

2. T ODAY ’ S PLAN Greeting Lesson: Navigation & the Protractor Postulate Homework / Questions Clean-up

3. P OINTS OF S AIL

4. B EARING / A ZIMUTH “The US Army defines the azimuth between Point A and Point B as the angle, measured in the clockwise direction, between the north reference ray and Ray AB. For example, if the bearing between Point A and Point B is E 45° S, the azimuth between Point A and Point B is 135°.” [2][3] azimuth [2][3]

5. M EASUREMENT OF A NGLES Definitions: A degree is 1/360 of the distance rotated in a circle. A rotation is the set of positions that are occupied by a ray as it turns 360°. A half-rotation is the set of positions for 180°.

6. A NGLE M EASURE : T HE P ROTRACTOR P OSTULATE The Protractor Postulate : It is possible to number the rays in a half-rotation with coordinates from 0 to 180, such that positive number differences measure angles. Note: This is the angle version of the Ruler Postulate. This means if rays OA and OB have coordinates a° and b°,  AOB = |a – b|. O A B

7. B ETWEENNESS OF R AYS Definition: A ray OB is between rays OA and OC (in the same half-rotation) iff its coordinate is between theirs. As with points (A-B-C), “OB is between OA and OC” is abbreviated “OA-OB-OC”. O A B C

8. A NGLE A DDITION : T HE B ETWEENNESS OF R AYS T HEOREM If OA-OB-OC, then  AOB +  BOC =  AOC. C O A B

9. F IELDS OF V ISION

10. M ETHODS OF BISECTION Compass and straightedge Ruler (for segments) or protractor (for angles) Paper folding

11. D EFINITIONS A point is the midpoint of a line segment iff it divides the segment into two equal segments. A line bisects an angle iff it divides the angle into two equal angles. Two things are congruent iff they would coincide exactly when superimposed: A B C D

12. D EFINITIONS, CONT. A corollary is a theorem that is easy (usually trivially easy) to prove as a consequence of another theorem or a postulate.

13. T WO B ASIC C OROLLARIES Corollary of the Ruler Postulate: A line segment has exactly one midpoint. Corollary of the Protractor Postulate: An angle has exactly one ray that bisects it.

14. H OMEWORK Asg #19: Read Ch. 3, Lesson 3 (pp. 93-97) & do: Exercises #1-7, 16-20, 34-43, 54 Bonus: Set III. Due Wednesday, 10/17. Test corrections are due Thursday (periods 1 & 2) or Friday (period 7). Asg #20: Read Ch. 3, Lesson 4 (pp. 100-103) & do: Set I, 1-6 and 10-14. Set II, 31-36, 41-47. Bonus: Set III. Due Monday, 10/22.

15. C LEAN - UP / R EMINDERS Pick up all trash / items. Push in chairs (at front and back tables). See you tomorrow!