Postulates are statements that are assumed to be true without proof. Postulates serve two purposes - to explain undefined terms, and to serve as a starting point for proving other statements.
Postulates of Equality
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
Point-Line-Plane Postulate
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.
��������������������������������������������������������������������������������������
