**Important Concepts and Formulas**

1. Axioms or postulates are the
assumptions which are obvious universal truths. They are not proved.

2. Theorems are statements which are
proved, using definitions, axioms, previously proved statements and deductive
reasoning.

3.

**Some of Euclid’s axioms were:**

i.
Things
which are equal to the same thing are equal to one another.

ii.
If
equals are added to equals, the wholes are equal.

iii.
If
equals are subtracted from equals, the remainders are equal.

iv.
Things
which coincide with one another are equal to one another.

v.
The
whole is greater than the part.

vi.
Things
which are double of the same things are equal to one another.

vii.
Things
which are halves of the same things are equal to one another.

4.

**Euclid’s postulates were:**

**Postulate 1:**A straight line may be drawn from any one point to any other point.

**Postulate 2:**A terminated line can be produced indefinitely.

**Postulate 3:**A circle can be drawn with any center and any radius.

**Postulate 4:**All right angles are equal to one another.

**Postulate 5:**If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

**5.**

**Two equivalent versions of Euclid’s fifth postulate are:**

i.
For
every line

*l*and for every point P not lying on*l*, there exists a unique line*m*passing through P and parallel to*l*.
ii.
Two
distinct intersecting lines cannot be parallel to the same line.

6. All the attempts to prove Euclid’s
fifth postulate using the first 4 postulate failed. But they led to the
discovery of several other geometries, called non-Euclidean geometries.