Geometry
Postulates & Theorems
Incidence Postulates
Postulate 1.1:
Expansion Postulate
A line contains at least two points. A plane contains at least three
noncollinear points. Space contains at least four noncoplanar points.
Postulate 1.2:
Line Postulate
Any two points in space line in exactly one line.
Postulate 1.3:
Plane Postulate
Three distinct noncollinear points lie in exactly one plane.
Postulate 1.4:
Flat Plane Postulate
If two points line in a plane, then the line containing these two points lies in
the same plane
Postulate 1.5:
Plane Intersection Postulate
If two planes intersect, then their intersection is exactly one line.
Incidence Theorems
Theorem 1.1
If two distinct lines intersect, they intersect in one and only one point.
Theorem 1.2
A line and a point not on that line are contained in one and only one plane.
Theorem 1.3
Two interesting lines are contained in one and only one plane.
Theorem 1.4
Two parallel lines are contained in one and only one plane.
Postulate 2.1:
Line Separation Postulate
Every point divides any line through that point into three disjoint sets: the
point and two-half lines
Postulate 2.2:
Plane Separation Postulate:
Every line divides any plane containing the line into three disjoint sets: the
line and two half-planes
Theorem 2.1:
Jordan Curve Theorem
Any simple closed curve divides a plane into three disjoint sets: the curve
itself, its interior, and its exterior