This document discusses Euclidean geometry. It begins by providing background on the origins of geometry in ancient Egypt and Greece. It then outlines Euclid's key contributions, including his definitions, axioms, and postulates laid out in his influential work "The Elements". Specific examples of Euclid's definitions, axioms like things equal to the same thing being equal, and postulates such as lines intersecting to form angles less than two right angles are described. Finally, it provides an example theorem and proof that two distinct lines cannot share more than one point.

1. PRESENTED BY:-
SIMRAN KHIRBAT
IX A

3. TABLE OF CONTENT
•Introduction
•Euclid’s Definition
•Euclid’s Axioms
•Euclid’s Five Postulates
•Theorems with Proof

4.  The word ‘Geometry’ comes from Greek word ‘geo’
meaning the ‘earth’ and ‘metrein’ meaning to
‘measure’. Geometry appears to have originated from
the need for measuring land.
 Nearly 5000 years ago geometry originated in Egypt
as an art of earth measurement. Egyptian geometry
was the statements of results.
 The knowledge of geometry passed from Egyptians to
the Greeks and many Greek mathematicians worked
on geometry. The Greeks developed geometry in a
systematic manner.

5.  Euclid was the first Greek Mathematician who
initiated a new way of thinking the study of
geometry.
 He introduced the method of proving a geometrical
result by deductive reasoning based upon previously
proved result and some self evident specific
assumptions called AXIOMS.
 The geometry of plane figure is known as ‘
Euclidean Geometry ’. Euclid is known as the father
of geometry.
 His work is found in Thirteen books called ‘ The
Elements ’.

6. Some of the definitions made by Euclid in
volume I of ‘The Elements’ that we take for
granted today are as follows :-
 A point is that which has no part
 A line is breadth less length
 The ends of a line are points
 A straight line is that which has length only

7.  The edges of a surface are lines
 A plane surface is a surface which lies evenly
with the straight lines on itself
o Axioms or postulates are the assumptions which
are obvious universal truths. They are not proved.
o Theorems are statements which are proved, using
definitions, axioms, previously proved statements
and deductive reasoning .

8. SOME OF EUCLID’S AXIOMS WERE :-
 Things which are equal to the same thing are
equal to one another.
i.e. if a=c and b=c then a=b.
Here a, b and c are same kind of things.
 If equals are added to equals, the wholes are
equal.
i.e. if a=b and c=d, then a+c = b+d
Also a=b then this implies that a+c = b+c .

9.  If equals are subtracted, the remainders are equal.
 Things which coincide with one another are equal
to one another.
 Things which are double of the same things are
equal to one another

10.  The whole is greater than the part.
That is if a > b then there exists c such that
a =b + c.
Here, b is a part of a and therefore, a is greater
than b.
 Things which are halves of the same things are
equal to one another.

11. EUCLID’S POSTULATES WERE :-
 POSTULATE 1 :-
 A straight line may be drawn from any one point
to any other point
 Axiom :-
 Given two distinct points, there is a unique line
that passes through them

12.  POSTULATE 2 :-
 A terminated line can be produced infinitely
 POSTULATE 3 :-
 A circle can be drawn with any centre and any radius
 POSTULATE 4 :-
 All right angles are equal to one another

13.  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.

14.  THEOREM :-
 Two distinct lines cannot have more than
one point in common
 PROOF :-
 Two lines ‘l’ and ‘m’ are given. We need
to prove that they have only one point in common
 Let us suppose that the two lines intersects
in two distinct points, say P and Q

15.  That is two line passes through two distinct points
P and Q
 But this assumptions clashes with the axiom that
only one line can pass through two distinct points
 Therefore the assumption that two lines intersect in
two distinct points is wrong
 Therefore we conclude that two distinct lines cannot
have more than one point in common