Through any two points there exists exactly one line. (Two points determine exactly one line.)
Through any three noncollinear points there exists exactly one plane. (Three noncollinear points determine exactly one plane.)
Flatness (Flat Plane) Postulate
If two points of a line lie on a plane, then the whole line lies on the plane.
To each set of two points there is a unique positive number called the distance between the two points
Given a line and the set of reals there is a one-to-one correspondence between the set of reals and the points on the line such that: if the coordinates of the points are a and b, then AB equals the absolute value of a minus b equals the absolute value of b minus a
Segment Addition Postulate
If R,P, and Q are three points on a line and R is a point between points P and Q, then PR plus RQ equals PQ
If R,P, and Q are three collinear points and PR plus RQ equals PQ, then R lies between P and Q
Angle Measurement Postulate
To each angle there corresponds a unique, positive number between zero degrees and one eighty degrees called its measure
given angle ACB and the coordinate of ray CA is "a" and the coordinate of ray CB is "b" then the measure of angle ACB equals the absolute value of a minus b
Angle Construction Postulate
Given a ray AB and one halfplane formed by line AB and a number x such that zero degrees is less than x is less than one eighty degrees, then there exists exactly one ray AC such that the measure of angle BAC equals x.
Angle Addition Postulate
If S is a point in the interior of angle PQR, then the measure of angle PQS plus the measure of angle SQR equals the measure of angle PQR
Plane Separation Postulate
Given a line and a plane containing it, the points of the plane not on the line form two nonempty, disjoint sets such that: 1)each set is convex and is called a halfplane 2)if P is in one halfplane and Q is in the other, then segment PQ intersects the given line or edge
Space Separation Postulate
Given space, the points of space that do not lie on a given plane form two nonempty, disjoint sets such that: 1)each set is convex and is called a halfspace 2)if P is in one halfspace and Q is in the other, then segment PQ intersects the given plane or face
Side Angle Side Postulate
Given a correspondence between triangles such that two pairs of corresponding sides are congruent and their included angles are congruent, then the triangles are congruent
Given a line and a point not on it, then there is exactly one line through the point parallel to the given line in the given plane.
Area Addition Postulate
If a region R is composed of two non-overlapping regions, then R equals Region one union Region two and the area of R equals the area of Region one plus the area of Region two.
The area of square region is side times side or the area of square region is side squared.