(A print-out present from Fabian)

Postulates and Theorems

A) Unique Line Assumption: Through any two points, there is exactly one line.
Note: This doesn't apply to nodes or dots.
B) Dimension Assumption: Given a line in a plane, there exists a point in the plane not on that line. Given a plane in space, there exists a line or a point in space not on that plane.
C) Number Line Assumption: Every line is a set of points that can be put into a one-to-one correspondence with real numbers, with any point on it corresponding to zero and any other point corresponding to one.
Note: This doesn't apply to nodes or dots. This was once called the Ruler Postulate.
D) Distance Assumption: On a number line, there is a unique distance between two points.
E) If two points lie on a plane, the line containing them also lies on the plane.
F) Through three noncolinear points, there is exactly one plane.
G) If two different planes have a point in common, then their intersection is a line.

Euclid's Postulates

A) Two points determine a line segment.
B) A line segment can be extended indefinitely along a line.
C) A circle can be drawn with a center and any radius.
D) All right angles are congruent.
E) If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180 degrees, then the lines will intersect on that side of the transversal.

Polygon Inequality Postulates

Triangle Inequality Postulate: The sum of the lengths of two sides of any triangle is greater than the length of the third side.

Quadrilateral Inequality Postulate: The sum of the lengths of 3 sides of any quadrilateral is greater than the length of the fourth side.

Theorems

Euclid's First Theorem

Line Intersection Theorem
Two different lines intersect in at most one point.

Betweenness Theorem
If C is between A and B and on , then AC + CB = AB.

Related Theorems:

Theorem:
If A, B, and C are distinct points and AC + CB = AB, then C lies on .

Theorem:
For any points A, B, and C, AC + CB

Pythagorean Theorem:
a² + b² = c², if c is the hypotenuse.

Right Angle Congruence Theorem:
All right angles are congruent.

Reflexive Property of Equality: a = a

Symmetric Property of Equality: if a = b, then b = a

Transitive Property of Equality: if a = b and b = c, then a = c .

Postulates of Equality and Operations

Addition Property of Equality: if a = b, then a + c = b + c

Multiplication Property of Equality: if a = b, then a * c = b * c

Substitution Property of Equality: if a = b, then a can be substituted for b in any equation or inequality

Subtraction Property of Equality: if a = b, then a - c = b - c

Postulates of Inequality and Operations

Addition Property of Inequality: if a < > b, then a + c < > b + c

Multiplication Property of Inequality: if a < b and c > 0, then a * c < b * c
if a < b and c < 0, then a * c > b * c

Equation to Inequality Property: if a and b are positive, and a + b = c, then c > a and c > b
if a and b are negative, and a + b = c, then c < a and c < b

Subtraction Property of Inequality: if a < > b, then a - c < > b - c

Transitive Property of Inequality: if a < b and b < c, then a < c

Postulates of Operation

Commutative Property of Addition: a + b = b + a

Commutative Property of Multiplication: a * b = b * a

Distributive Property: a * (b + c) = a * b + a * c and vice versa