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Unformatted text preview: 26 Betweenness, Segments and Rays
A) Between a. Example of between: b. Example of not between: c. Definition i. B is between A and C if 1. A, B, and C are different points of the same line, and 2. AB + BC = AC. B) Theorem 23 a. Let A, B, and C be points of a line, with coordinates x, y, and z respectively. If x . y . z , then ABC. b. Example: C) Draw a line with three points. a. One point is always _________ b. Theorem 24 i. If A, B, and C are three different points of the same line, then exactly one of them is between the other two. D) Postulate 4: The Line Postulate a. For every two different points there is exactly one line that contains both points i. However you draw two points, there is one that goes thru them ii. The shortest distance between two points is a straight line\ iii. Notation: E) Segment a. For any two points A and B, the segment AB is the union of A and B, and all points that are between A and B. i.
ii. iii. The points A and B are called the end points of AB Notation: F) Ray (1) AB and (2) the set of all points C for which ABC. The point A is called the end point of AB . b. Notation:
a. Let A and B be points. The ray AB is the union of G) Opposite Ray If A is between B and C, then AB and rays H) Draw points C and D
a. AC are called opposite a. Now construct a line: b. A segment: c. A ray: d. A different ray: e. Opposite rays with point M: I) Theorem 25. The PointPlotting Theorem a. 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. J) Midpoint a. A point is called a midpoint of a segment C and AB = BC AC if B is between A and K) Theorem 26. The Midpoint Theorem a. Every segment has exactly one midpoint L) Bisector a. The midpoint of a segment AB , or any line, plane, ray, or segment which contains the midpoint and does not contain AB HW# 5 P. 42 125 ...
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 Spring '09
 Johnson
 Algebra

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