MATHEMATICS PROJECT
The students of class IX-A classavnit kaur, mehakdeep kaur,
amanpreet kaur, gurleen kaur, manimahesh, japtyesh singh, harleen
kaur, gurprince brar, simrandeep singh, amandeep singh, navkiran
kaur, sukhmandeep singh
,MATHEMATICS PPTPROJECT
presented BY :-
INTRODUCTION TO EUCLID'S GEOMETRY
TABLE OF CONTENT Introduction Euclids Definition Euclids Axioms
Euclids 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 geometryHe introduced the method of
proving a geometrical result by deductive reasoning based upon
previously proved result and some self evident specific assumptions
called AXIOMSThe 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.
EUCLIDS 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.
EUCLIDS AXIOMsSOME OF EUCLIDS 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.
EUCLIDS FIVE POSTULATESEUCLIDS 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.
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..THEOREM Two distinct lines cannot have more than one
point in common PROOFTwo lines l and m are given. We need to prove
that they have only one point in commonLet 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
Thank You
THE END..
