presented BY :-}

01. SHARMILA KRISHNASWAMY Ix b class

Jnv shivaraguddaMandya

1.

INTRODUCTION TO

EUCLID'S GEOMETRY

TABLE OF CONTENT IntroductionEuclid’s Definition Euclid’s Axioms Euclid’s Five Postulates Theorems with Proof

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.

.

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’.

EUCLID’S DEFINITONSSome 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 partA line is breadth less lengthThe ends of a line are pointsA straight line is that which has length

only

Continued…..The edges of a surface are linesA plane surface is a surface which lies

evenly with the straight lines on itselfAxioms 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.

EUCLID’S AXIOMsSOME 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.

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.

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.

EUCLID’S FIVE POSTULATESEUCLID’S POSTULATES WERE :-

POSTULATE 1:- A straight line may be drawn from any one

point to any other pointAxiom :-Given two distinct points, there is a unique line

that passes through them

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

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.

ExampleExample :- In fig :- 01 the line EF falls In fig :- 01 the line EF falls on two lines AB and CD such that the on two lines AB and CD such that the angle m + angle n < 180° on the angle m + angle n < 180° on the right side of EF, then the line right side of EF, then the line eventually intersect on the right side eventually intersect on the right side of EFof EF

fig :- o1

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

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

THE END