Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus.

A conjecture is an unproven statement that is based on observations.

Inductive reasoning is a process that involves looking for patterns and making conjectures.

A counterexample is an example that shows a conjecture is false. 

A point has no dimension, a line extends in one dimension, and a plane extends in two dimensions.

Collinear points are points that lie on the same line.

Coplanar points
are points that lie on the same plane.

On a line passing through points A and B, segment AB consists of all points between A and B and endpoints A and B.

On a line passing through points A and B, ray AB consists of the initial point A and all points on the same side of A as point B.

If point C is between A and B, then ray CA and ray CB are opposite rays.

Two or more geometric figures intersect if they have one or more points in common.

The intersection of the figures is the set of points the figures have in common. 

A postulate or axiom is a rule that is accepted without proof.

Postulate 1 Ruler Postulate:

The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point.

The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B.

AB is also called the length of AB.

When three points lie on a line, you can say that one of them is between

the other two.

Postulate 2 Segment Addition Postulate:

If B is between A and C, then AB + BC AC. If AB +BC = AC, then B is between A and C.

The Distance Formula is a formula for computing the difference between two points in a coordinate plane.

The Distance Formula:

If A(x1, y1and B(x2, y2)are points in a coordinate plane, then the distance between A and B is

Segments that have the same length are called congruent segments.