The Basic Postulates & Theorems of Geometry
These are the basics when it comes to postulates and theorems in Geometry. These are the ones that you have to know.
Postulates
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.
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.
Note: This part has been proven as a theorem. See below, proof.
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
Theorums are statements that can be deduced and proved from definitions, postulates, and previously proved theorums.
Euclid's First Theorem: The triangle in the picture is an equilateral triangle.
Note: D. Joyce's Elements page (the link above) is where you'll find anything else you need to know about Euclid's ideas, postulates, and theorems.
Line Intersection Theorem: Two different lines intersect in at most one point. For proof see Unique Line Assumption
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. See proof.
Note: While you can usually get away with not knowing the names of theorems, your Geometry teacher will generally require you to know them.
Algebra Postulates
Here are the basic postulates of equality, inequality, and operations. Dave didn't get a chance to write them, and I needed them for my section on the basic postulates of Geometry (review is always good). Have a blast!
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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
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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
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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