GEOMETRY Legal Reasons
Level 1
Lines, Segments, Rays, Distance
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1

Point-Line-Plane Postulate

1. Unique Line Assumption: through any two points there is exactly one line.
2. Number Line Assumption: Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.
3. Dimension Assumption:
   1.) Given a line in a plane, there is at least one point in the plane that is not on the line.
   2.) Given a plane in space, there is at least one point in space that is not in the plane.

2

Line Intersection Theorem

Two different lines intersect in at most one point.

3

Definition of Parallel Lines

Two coplanar lines s and t are parallel lines, written s//t, if and only if they have no points in common or they are identical. 

4

Definition of Betweenness

A point is between two other points if it is on the same line and its coordinate is between their coordinates.

5

Definition of Segment

The segment (or line segment) with endpoints A and B, denoted is the set consisting of the distinct points A and B and all points between A and B.

6

Definition of Ray

The ray with endpoint A and containing a second point B, denoted by , consists of the points on and all points for which B is between each of them and A.

7

Definition of opposite rays

8

The Distance Postulate

1. Uniqueness Property: On a line, there is a unique distance between two points.
2. Distance Formula: If two points on a line have coordinates x and y, the distance between them is
   |x - y|.
3. Additive Property: If B is on , then
   AB + BC = AC