## Postulates

1. Given any two points, there is exactly one line which contains both of them.
2. The Distance Postulate: Given any pair of distinct points, there corresponds a unique positive real number called the distance between the two points.
3. The Ruler Postulate: The points of a line can be placed in correspondence in such a way that:
• there is a one to one correspondence between the set of points on the line and the set of real numbers, and
• the distance between two points equals the absolute value of the difference of the corrsponding numbers.
4.
1. Every plane contains at least three noncollinear points.
2. Every space contians at least foud noncoplanar points.
3. Hyperspace contains at least five noncospatial points

5. If a plane contains two points of a line, then the plane contains the whole line.
6. If a space contains three noncollinear points of a plane, then it contains the whole plane.
7. Any three points lie in at least one plane, and any three noncollinear points lie in exactly one plane.
8. Any four points lie in at least one space, and any four noncoplanar points lie in exactly one space.
9. If two distinct spaces intersect, then their intersection is a plane.
10. The Plane Separation Postulate: Given any line and any plane containing it. The points of the plane that do not lie on the line form two disjoint sets such that
1. each of the sets is convex and non-empty, and
2. if P is any point of one set and Q is any point in the other set, then the segment P--Q intersects the line.

11. The Space Separation Postulate: Given any plane and any space containing it. The points of the space that do not lie on the plane form two disjoint sets such that
1. each of the sets is convex and non-empty, and
2. if P is any point of one set and Q is any point in the other set, then the segment P--Q intersects the plane.

12. The Hyperspace Separation Postulate: The points of hyperspace that do not lie in a given space form two disjoint sets such that
1. each of the sets is convex and non-empty, and
2. if P is any point of one set and Q is any point in the other set, then the segment P--Q intersects the space.

## Theorems

2.1. The Ruler Placement Theorem:Given any two points P and Q of a line, there is a coordinate system for the line such that the coordinate of P is zero and the coordinate of Q is positive.
2.2. Let A, B, and C be three distinct points of a line, with the respective coordinates x, y, and z. If x < y < z, then A-B-C.
2.3. Given any three distinct points on a line, exactly one is between the other two.
2.4. Given any coordinate system for a line. Of any two opposite rays on the line, one ray is the set of all points having coodinates greater than or equal to the coordinate of the common endpoint, and the other ray is the set of all points having coordinates less than or equal to the coordinate of the common endpoint.
2.5. The Point-Plotting Theorem: Let A->B be a ray and let x be a positive number. Then there is exactly one point P of A->B such that AP = x.
2.6. The Midpoint Theorem: Every segment has exactly one midpoint.
3.1. If two lines intersect, then their intersection contains exactly one point.
3.2. If a line intersects a plane not containing it, then the intersection contains exactly one point.
3.3. Ommitted.
3.4. Let l be a line and P be a point not on l. Then there is exactly one plane containing l and P.
3.5. Let l1 and l2 be any two intersecting lines. Then there is exactly one plane containing them.
3.6. Let E be a plane and P a point not on it. Then there is exactly one space containing E and P.
3.7. Let E and F be two intersecting planes. Then there is exactly one space containing them.
3.8. Given any plane E, any line l in E, and any two points A and B in E but not on l. If no point of l is between A and B, then A and B lie in the same half-plane with edge l.
3.9 Given any plane E, any line l in E, and any two points A and B that are in E but not on l. If some point of l is between A and B, then A and B are in opposite half-planes with edge l.
3.10 Given any space S and any plane E in S and any two points A and B that are in S but not on E. If no point of E is between A and B, then A and B are in the same half-space with face E.
3.11 Given any space S and any plane E in S, and any two ponts A and B that arein S but not on E. If some point of E is between A and B, then A and B are in opposite half-spaces with face E.
3.12 If a ray contains a point of a half-plane, and if the endpoint of the ray is on the edge of the half-plane, then the interior of the ray lies in the half-plane.
3.13 If a ray contains a point of a half-space, and if the endpoint of the ray is on the face of the half-space, then the interior of the ray lies in the half-space.
3.14 If a ray contains a point of a half-hperspace, and if the endpoint of the ray is on the face of the half-hyperspace, then theinterior of the ray lies in the half-hyperspace.