# Geometry/The SMSG Postulates for Euclidean Geometry

SMSG Postulates

Undefined Terms:

1. Point
2. Straight line (for simplicity just called "line")
3. Plane
1. Postulate 1. (Line Uniqueness) Given any two different points, there is exactly one line which contains both of them.
2. Postulate 2. (Distance Postulate) To every pair of different points there corresponds a unique positive number.
3. Postulate 3. (Ruler Postulate) The points of a line can be placed in correspondence with the real numbers in such a way that
1. To every point of the line there corresponds exactly one real number called the point’s coordinate,
2. to every real number there corresponds exactly one point of the line, and
3. The distance between two points is the absolute value of the difference of the corresponding coordinates.
4. Postulate 4. (Ruler Placement Postulate) Given any two points $P$ and $Q$ on a line, the coordinate system can be chosen in such a way that the coordinate of $P$ is zero and the corrdinate of $Q$ is positive.
5. Postulate 5. (Points Exist) (a) Every plane contains at least three non-collinear points. (b) Space contains at least four non-coplanar points.
6. Postulate 6. (Points) If two points lie in a plane, then the whole of the line containing these points lies in the same plane.
7. Postulate 7. (Plane Uniqueness) There is at least one plane containing any three points, and exactly one plane containing any three non-collinear points.
8. Postulate 8. (Plane Intersection) If two different planes intersect, then their intersection is a line.
9. Postulate 9. (Plane Separation Postulate) Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that
1. each of the sets is convex, and
2. if P is in one set and Q is in the other, then the segment ${\displaystyle {\overline {PQ}}}$ intersects the line.
10. Postulate 10. (Space Separation Postulate) The points of space that do not lie in a given plane form two sets such that
1. each of the sets is convex, and
2. if P is in one set and Q is in the other, then the segment ${\displaystyle {\overline {PQ}}}$ intersects the plane.
11. Postulate 11. (Angle Measurement Postulate) To every angle there corresponds a real number between 0° and 180°.
12. Postulate 12. (Angle Construction Postulate) Let ${\displaystyle {\overrightarrow {AB}}}$ be a ray on the edge of the half-plane $H$. For every $r$ between 0 and 180 there is exactly one ray ${\displaystyle {\overrightarrow {AP}}}$ with P in H such that ${\displaystyle m\angle PAB=r}$.
13. Postulate 13. (Angle Addition Postulate) If D is a point in the interior of ${\displaystyle m\angle BAC}$ , then $m\angle BAC = m\angle BAD + m\angle DAC$.}
14. Postulate 14. (Supplementary Postulate) If two angles form a linear pair, then they are supplementary.
15. Postulate 15. (Side Angle Side Postulate) Given a one-to-one correspondence between two triangles (or between a triangle and itself), if two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, the correspondence is a congruence.
16. Postulate 16. (Parallel Postulate) Through a given external point there is at most one line parallel to a given line.
17. Postulate 17. To every polygonal region there corresponds a unique positive real number called its area.
18. Postulate 18. If two triangles are congruent, then the triangular regions have the same area.
19. Postulate 19. Suppose that the region $R$ is the union of two regions $R_1$ and $R_2$. If $R_1$ and $R_2$ intersect in at most a finite number of segments and points, then the area of $R$ is the sum of the areas of $R_1$ and $R_2$.
20. Postulate 20. The area of a rectangle is the product of the length of its base and the length of its altitude.
21. Postulate 21. The volume of a rectangular parallelepiped is equal to the product of the length of its altitude and the area of the base.
22. Postulate 22. (Cavalieri’s Principle) Given two solids and a plane, if for every plane that intersects the solids and is parallel to the given plane the two intersections determine regions that have the same area, then the two solids have the same volume.