1. Betweenness, Segments and Rays, Point-Plotting Thereom
Proof Geometry

2. Betweenness B is between A and C if:
1) A, B, and C are different points of the same line, and
2) AB + BC = AC.
When B is between A and C, we write: A-B-C or C-B-A

3. The Line Postulate
How many different lines can be drawn between two points?
For every two different points, there is exactly ONE line that contains both points

4. Segments and Rays Three line segments exist in the picture. Name them
Four Rays exist. Name them.
What is the difference between a segment and a ray?

5. Formal definition of segment
For any two points A and B, the segment 𝐴𝐵 is the union of A, B, and all points that are between A and B.
AB is the length of 𝐴𝐵

6. Formal Definition of Ray
Let A and B be points. The ray 𝑨𝑩 is the union of:
i.) 𝐴𝐵
ii.) The set of all points C for which A*B*C.
A-B-C

7. Opposite Rays What are two opposite rays?
Formal Definition: If A is between B and C then are called opposite rays.

8. Point-Plotting Theorem
Let 𝐴𝐵 be a ray, and let x be a positive number. Then there is exactly one point P of 𝐴𝐵 such that AP = x.
Proof: x

9. Midpoint
Definition: A point B is called a midpoint of a segment 𝐴𝐶 if B is between A and C and AB = BC. We say B bisects 𝐴𝐵 .
Ex) If AB = x+2, and BC = 2x-2, What is AC?

10. The Midpoint Theorem
Every segment has exactly one midpoint

11. Homework
P. 42 #1-4, 11-15, 18, 19