# Geometry

closurea + b is unique real number
commutativea + b = b + a
associative(a + b) + c = a + (b + c)
additive of zeroa + 0 = a and 0 + a = a
additive inversesa + (-a) = 0 and (-a) + a = 0

## Multiplication Properties

closurea * b is a unique real number
commutativeab = ba
associative(ab)c = a(bc)
multiplication postulate of onea * 1 = a and 1 * a = a
multiplicative inversesa * 1/a = 1
distributivea(b + c) = ab + ac and
(b + c)a = ba + ca

## Equality and Inequality

reflexivea = a
symmetricif a = b, b = a
transitiveif a = b and b = c, then a = c
comparisononly one is true:
a < b, a = b, b < a
transitive postulateif a < b and b < c, then a < c
additive postulateif a < b, then a + c < b + c
multiplicative postulateif a < b and 0 < c,
then ac < bc;
if a < b and c < 0,
then bc < ac

## Proved Properties (Theorems)

Addition (equality)if a = b, then
a + c = b + c and
c + a = c + b
Subtraction (equality)if a = b, then
a - c = b - c and
c - a = c - b
Multiplication (equality)if a = b then ac = bc
Division (equality)if a = b, and c ≠ 0,
then a/c = b/c
Subtraction (inequality)if a < b, then
a - c < b - c and
c - a > c - b
Division (inequality)if a < b, and c > 0,
then a/c < b/c;
if a < b, and c < 0,
then a/c > b/c
Substitution Principleif a = b, a can be replaced by b
Zero-Product Propertyif ab = 0, then a = 0 or b = 0

## Postulates

P1A line contains at least two points.
A plane contains at least three points (not all on one line).
Space contains at least 4 points (not all in one plane).
P2Through any two different points, there is exactly one line.
P3Through any three points (which are not on one line), there is exactly one plane.
P4If two points lie in a plane,
then the line containing them lies in that plane.
P5If two different planes intersect,
then their intersection is a line.
P6Between any 2 points there is a unique distance.
P7(Ruler Postulate) AB= lx-yl and there is a one-to-one correspondence with all real numbers and points on the number line.