presented BY :-

Shashwat Jha class -Ix g Roll no. -44

subject teacher- Aparna ma’am

Content listTABLE OF CONTENT

Introduction

Euclid’s Definition

Euclid’s Axioms

Euclid’s Five Postulates

Theorems with Proof

1.

INTRODUCTION

Nearly 5000 years ago geometry originated in Egypt as an art of earth measurement. Egyptian geometry was the statements of results.

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.

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.

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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 DEFINITONS

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

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.

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be

created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

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.

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.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

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

Presented by-Shashwat Jha