# 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. See construction/proof at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
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.

- Jaime III
"Not failure, but low aim, is a crime." - James Russel Lowell