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

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

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

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?

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

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

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

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

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?

The Midpoint Theorem Every segment has exactly one midpoint

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