Euclid's Geometry

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presented BY :-} 01. SHARMILA KRISHNASWAMY Ix b class Jnv shivaragudda Mandya FA 2 MATHEMATICS PROJECT :- 01

INTRODUCTION TO EUCLID'S GEOMETRY :

INTRODUCTION TO EUCLID'S GEOMETRY

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

INTRODUCTION :

INTRODUCTION The word ‘ Geometry ’ comes from Greek words ‘ geo ’ meaning the ‘ earth ’ and ‘ metrein ’ meaning to ‘ measur e’. 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. .

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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:

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

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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.

EUCLID’S AXIOMs:

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.

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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.

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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 POSTULATES:

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

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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

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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.

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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

CONTINUED…..:

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

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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

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THE  END

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