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Proof:

Given: AB and CD are two lines.

To prove:

They intersect at one point or they do not intersect.

Proof:

Suppose the lines AB and CD intersect at two points P and Q.

This implies the line AB passes through the points P and Q.

Also the line CD passes through the points P and Q.

This implies there are two lines which pass through two distinct point P and Q. But we know that one and only one line can pass through two distinct points. This axiom contradicts out assumption that two distinct lines can have more than one point in common.

The lines AB and CD cannot pass through two distinct point P and Q.

The statement or results which were proved by using Euclid's axioms and postulates are called propositions or Theorems.

Euclid listed 23 definitions in book 1 of the 'elements'. We list a few of them:

1) A point is that which has no part

2) A line is a breadth less length

3) The ends of a line are points

4) A straight line is a line which lies evenly with the points on itself.

5) A surface is that which has length and breadth only.

6) The edges of a surface are lines

7) A plane surface is surface which lies evenly with straight lines on its self.

Starting with these definitions, Euclid assumed certain assumptions, known as axioms and postulates.

A system of axioms is said to be consistent, if it is impossible to deduce a statement from these axioms, which contradicts any of the given axioms or proposition.