Euclid was a Greek mathematician from Alexandria known as the "Father of Geometry". In his influential work Elements, he deduced the principles of Euclidean geometry from 5 postulates (axioms) for plane geometry related to drawing lines and circles. The postulates state that a line can be drawn between any two points, a line can be extended indefinitely, a circle can be drawn with any center and radius, all right angles are equal, and if two lines intersect another such that the interior angles on the same side sum to less than two right angles, the two lines will intersect on that side. Euclid's work was foundational and served as the main geometry textbook for over 2000 years.

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EUCLID’S
POSTULATE

2. EUCLIDEUCLID
Euclid (300 BC), also known as Euclid of Alexandria,Euclid (300 BC), also known as Euclid of Alexandria,
was a Greek mathematician, often referred to as thewas a Greek mathematician, often referred to as the
"Father of Geometry". He was active in the Greek"Father of Geometry". He was active in the Greek
colony of Alexandria. during the reign of Ptolemy Icolony of Alexandria. during the reign of Ptolemy I
(323–283 BC). His(323–283 BC). His ElementsElements is one of the mostis one of the most
influential works in the history of mathematics,influential works in the history of mathematics,
serving as the main textbook for teachingserving as the main textbook for teaching
mathematics (especially geometry) from the time ofmathematics (especially geometry) from the time of
its publication until the late 19th or early 20thits publication until the late 19th or early 20th
century. In thecentury. In the ElementsElements, Euclid deduced the, Euclid deduced the
principles of what is now called Euclidean geometryprinciples of what is now called Euclidean geometry
from a small set of axioms. Euclid also wrote worksfrom a small set of axioms. Euclid also wrote works
on perspective, conic sections, spherical geometry,on perspective, conic sections, spherical geometry,
number theory and rigor. Euclid's contributions werenumber theory and rigor. Euclid's contributions were
built on the Greek culture of philosophicalbuilt on the Greek culture of philosophical
investigation into which he was born, relying on theinvestigation into which he was born, relying on the
works of thinkers such as Plato, Aristotle, Eudoxus,works of thinkers such as Plato, Aristotle, Eudoxus,
and Theaetetus.and Theaetetus.
EUCLIDEUCLID

7. AXIOMSAXIOMS
In mathematics, the termIn mathematics, the term axiomaxiom is used in twois used in two
related but distinguishable senses: "logical axioms“related but distinguishable senses: "logical axioms“
and "non-logical axioms". In both senses, an axiom isand "non-logical axioms". In both senses, an axiom is
any mathematical statement that serves as aany mathematical statement that serves as a
starting point from which other statements arestarting point from which other statements are
logically derived. Unlike theorems, axioms (unlesslogically derived. Unlike theorems, axioms (unless
redundant) cannot be derived by principles ofredundant) cannot be derived by principles of
deduction, nor are they demonstrable by mathematicaldeduction, nor are they demonstrable by mathematical
proofs, simply because they are starting points; thereproofs, simply because they are starting points; there
is nothing else from which they logically followis nothing else from which they logically follow
(otherwise they would be classified as theorems).(otherwise they would be classified as theorems).

8. POSTULATESPOSTULATES
Euclidean geometry is an axiomatic system, in which allEuclidean geometry is an axiomatic system, in which all
theorems ("true statements") are derived from a smalltheorems ("true statements") are derived from a small
number of axioms. Near the beginning of the first book ofnumber of axioms. Near the beginning of the first book of
the Elements, Euclid gives fivethe Elements, Euclid gives five postulatespostulates(axioms) for plane(axioms) for plane
geometry,stated in terms of constructions:geometry,stated in terms of constructions:
11stst
POSTULATEPOSTULATE ::
A straight line may be drawn from any one point to any
other point.Same may be stated as axiom .Given two
distinct points, there is a unique line that passes through
them.

9. This postulates simple says that if you have any twoThis postulates simple says that if you have any two
points--A and B, say--then you can always connectpoints--A and B, say--then you can always connect
them with a straight line.them with a straight line.
22NDND
POSTULATE:POSTULATE:
A terminated line can be produced indefinitely.
It tells us that we can always make a line segment
longer. That means that we never run out of space;
that is, space is infinite.

10. 3RD
POSTULATE:
A circle can be drawn with any centre and any radius.
It allows for the existence of circles of any size and
center--say center A and radius AB.

11. 44THTH
POSTULATE:POSTULATE:
All right angles are equal to one another.
It just says that whenever we create a right angle by
erecting perpendiculars, the angles so created are
always the same. Sameness here means that were we to
manipulate the angles by sliding them over the page,
they would coincide.

12. 5TH
POSTULATE:
If a straight line falling on two straight lines makes the
interior angle on the same side of it taken together less
than two right angles, then two straight lines, if produced
indefinitely, meet on that side on which the sum of the
angles is less than two right angles

13. The two different version of fifth postulate
a) For every line l and for every point P not lying on l, there
exist a unique line m passing through P and parallel to l.
b) Two distinct intersecting lines can not be parallel to the
same line
Equivalent Versions of Euclid's Fifth
Postulate
Although Euclid's statement of the postulates only
explicitly asserts the existence of the constructions, they
are also taken to be unique.
The Elements also include the following five "common
notions :

14. 1. Things that equal the same thing also
equal one another.
2.If equals are added to equals, then the
wholes are equal.
3.If equals are subtracted from equals, then the
remainders are equal.
4.Things that coincide with one another
equal one another.
5.The whole is greater than the part

15. Methods of Proof
Euclidean geometry is constructive.
Postulates 1, 2, 3, and 5 assert the existence
and uniqueness of certain geometric figures,
and these assertions are of a constructive
nature: that is, we are not only told that
certain things exist, but are also given
methods for creating them with no more
than a compass and an unmarked
straightedge.

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