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2. INTRODUCTION TO
EUCLID'S GEOMETRY

3. TABLE OFCONTENT
 Introduction
 Euclid’s Definition
 Euclid’s Axioms
 Euclid’s Five Postulates
Theorems with Proof

4. INTRODUCTION
 The word ‘Geometry’ comes from Greek words ‘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. EUCLID’S DEFINITONS
Some of the definitions made by Euclid involume I of
‘The Elements’ that we take for granted today are as follows :-
A point is that which has no part
A line is breadthless length
The ends of a line are points
A straight line is that whichhas lengthonly

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

8. EUCLID’S AXIOMs
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.

9. Continued…..
i.e. if a=b and c=d, then a+c = b+d
Also a=b then this implies that a+c=b+c.
If equals are subtracted, the remainders are equal.
Things which coincide with one another are equal to one
another.

10. Continued…..
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 double of the same things are equal to one
another.
 Things which are halves of the same things are equal to one
another.

11. EUCLID’S FIVE POSTULATES
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. Continued…..
 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. Continued…..
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. Example :-
 In fig :- 01 the line EF falls on two lines AB and CD such that
the angle m + angle n < 180° on the right side of EF, then the
line eventually intersect on the right side of EF
fig :- o1

15. CONTINUED…..
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

16.  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