 Geometry rename
Updated 2009-02-23 18:45 edit

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 edit

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 edit

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 edit

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 edit

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. edit

Theorems

Theorem 3-1If 2 lines intersect, they intersect at exactly one point.
Thoerem 3-2If a point lies outside a line, exactly one plane contains the point and the line.
Thoerem 3-3If 2 lines intersect, only one plane contains both lines.
Thoerem 3-4On a ray there is exactly one point at a given distance from the ray's endpoint.
Thoerem 3-5A segment has exactly one mid-point.
Pythagorean Theorema2 + b2 = c2 in a right triangle when a and b are the legs and c is the hypotenuse.

See Geometry 2 and Math for more.