1. A NGLES AND P OSTULATES September 16, 2009

2. O BJECTIVES Content Objectives Students will review defined terms. Students will learn about angles and postulates. Language Objectives Students will share their answers using the smartboard. Students will show understanding of angles and postulates in writing.

7. A BOUT ANGLES An angle is the figure formed by two rays with the same endpoint. This shared endpoint is called the vertex. The two rays are called the sides of the angle. We can name the angle B, angle ABC, angle CBA, or angle 1.

8. N AMING ANGLES Find several ways to name the two angles.

9. M EASURING ANGLES Angles are measured in degrees. Like distance, degrees are positive. In geometry we measure angles in degrees and angles can have degree measures from 0° to 180°.

10. C LASSIFYING ANGLES Angles are classified according to their measures. Acute angle: Measure between 0 and 90 Right angle: Measure 90 Obtuse angle: Measure between 90 and 180 Straight angle: Measure 180

11. C ONGRUENT A NGLES AND A DJACENT A NGLES Congruent angles are angles that have equal measures. Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points. In essence it tells us that if the angles "overlap" then we cannot call them adjacent.

12. M AKING CONCLUSIONS Do not assume anything that a diagram does not tell you! Notice the marks for congruence in line segments and angles, as well as the mark for a 90 degree angle.

13. W HAT IS A POSTULATE ? Postulates or axioms are known as basic assumptions. Assumptions are statements accepted without proof.

14. P OSTULATE 1 R ULER P OSTULATE 1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1. 2. Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

15. P OSTULATE 2 S EGMENT A DDITION P OSTULATE If B is between A and C, then AB + BC = AC

16. P OSTULATE 3 P ROTRACTOR P OSTULATE On line AB in a given plane, choose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with the real numbers from zero to 180 in such a way that: a. OA is paired with 0, and OB with 180. b. If OP is paired with x, and OQ with y, then the measure of POQ = | x - y |.

17. P OSTULATE 4 A NGLE A DDITION P OSTULATE If point B lies in the interior of angle AOC, then the measure of angle AOB + the measure of angle BOC = the measure of angle AOC. If angle AOC is a straight angle and B is any point not on AC, then the measure of angle AOB + the measure of angle BOC = 180.

18. P OSTULATE 5 A line contains at least two points; a plane contains at least 3 points not all in one line; space contains at least four points not all in one plane.

19. P OSTULATE 6 Through any two points there is exactly one line. This is true for plane geometry.

20. P OSTULATE 7 Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.

21. P OSTULATE 8 If two points are in a plane, then the line that contains the points is in that plane.

22. P OSTULATE 9 If two planes intersect, then their intersection is a line.

23. D ID WE MEET OUR OBJECTIVES ? Content Objectives Students will review defined terms. Students will learn about angles and postulates. Language Objectives Students will share their answers using the smartboard. Students will show understanding of angles and postulates in writing.