Geometry Postulates and Important Properties

For use in proofs or solving problems.
The Postulates with titles in blue font are especially useful!

Postulates

Ruler Postulate (Ch. 1)	The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1. (We can arbitrarily set up any measurement system we like without changing the rules of geometry)
Segment Addition Postulate (Ch. 1)	If is between and , then (If two segments share an endpoint, B, then the segments can add together to make a longer segment.)
Protractor Postulate	On in a given plane, and any point on , all the rays originating at can be paired with real numbers to measure an angle. (We can measure angles however we want, though there are only two common systems for angle measure. We’ll use degrees in Geometry; an angle of 0° is the measure of the angle between two lines that share all their points in common. An angle of 180° is the angle formed by opposite rays on a straight line.)
Angle Addition Postulate (Ch. 1)	If point lies in the interior of , then the ray forms two new adjacent angles, and . Then, and . (Adjacent angles add together to form larger angles, and there measures also add the same way.)
Lines and Points (ch. 1)	A line contains at least two points; and through any two points there is exactly one line. (These two are converses of each other)
Points and Planes (Ch. 1)	A plane contains at least three points, and through any three [non-collinear] points there is exactly one plane.
Lines and Planes (Ch. 1)	If two planes intersect, their intersection is a single line. If two points are in a plane, then the line between those two points is also in the plane.
Parallel Postulate & its converse (Ch. 3)	If two parallel lines are intersected by a transversal line, then the corresponding angles formed are congruent. The converse is also true: If a transverse line intersects two other lines, and the corresponding angles formed are congruent, then the lines are parallel. (Simply put, when parallel lines exist, the angles formed by any third line are the same size where that line meets both the parallel lines.)

Properties of Equality
From Algebra and Geometry

Assume that are any real numbers.
The reflexive, symmetric, and transitive properties also apply to geometric figures.

Reflexive Property	(anything equals itself)
Symmetric Property	If then (It doesn’t matter what order we state an equality)
Transitive Property	If and , then . (If two things are both equal to a third thing, they must be equal to each other)
Addition/Subtraction Property	If then and (If you know two things are equal, they remain equal when you add or subtract the same thing to both)
Multiplication/Division Property	If then and . (If you know two things are equal, they remain equal when you multiply or divide them both by the same amount)
Associative Property	(The product of a sum is the same as the sum of the products.)