Class-IX Unit-5 Introduction to Euclid Geometry

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Euclid Geometry

• Euclid a Greek mathematician is called the Father of Geometry
• Euclid defined around 23 items. 7 of them are mentioned below

1) A point is that which has no part
2) A line is breath less and has length only
3) The end of a line is points
4) A straight line is a line which lies evenly with the points on itself
5) A surface is that which has length and breath only
6) The edges of a surface are lines

The definitions of line, point, plane explained by Euclid is not accepted by the Mathematician. So these terms are taken as undefined

Axioms or Postulates

Axioms or Postulates are assumptions which are obvious universal truths. They are not proved
Euclid axioms are the assumptions which are used throughout mathematics while Euclid Postulates are the assumptions which are specific to geometry

Statements

A sentence which is either True or falsebut but not both is called the statement.

Theorems

They are statements which are proved using axioms/postulates, definition, previously proved statement and deductive reasoning

Corollary

A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective

Euclid axioms

1) Things which are equal to same things are equal to one another

If x=z, y=z then x=y

2) If equals are added to equals, the wholes are equal

x=y => x+z=y+z

3) If equals are subtracted from equals, the remainders are equal

X=y => x-z=y-z

4) Things which coincide with one another are equal to one another

5) The whole is greater than the part

6) Things which are double of the same things are equal to one another

7) Things which are halves of the same things are equal to one another

8) If first thing is greater than second and second is greater than third, then first is greater than third

Euclid Postulates

1) A straight line may be dawn from one point to another point

2) A terminated line can be produced indefinitely

3) A circle can be drawn with any center and any radius

4) All right angles are equal to one another

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 straight lines if produced indefinitely meet on that side on which the angles are less than the two right angles

Play fair Axiom

For every line l and for every point P not lying on the line l, there exists a unique line m passing through P and Parallel to l

Some More important defination's

a) Point: It is denoted by a single dot on the paper and it is represented by english alpabet. It has no lenght ,breadth and thickness
b) Line: A line is straight and has no curves. It extends in both the direction and has no end point. We have infinate lines through a point while we will have only one line through two points
c) Line Segment:If we take a portion of a line ,then it is called line segment and it has lenght and two end points. It can be measured
d) Ray: If a line has one end point and it can extent in other direction,then it is called a ray
e) Congurent segments: If two line segments are equal then they are called congurent

Incident Axioms

1) A line contains infinitely many points
2) An infinte lines can be drawn throught a point
3) If we have two given points,then there exists only one line which have both the points on it
4) Three or more points are collinear if they line on the same line
5) Two or more lines are concurrent lines if there is a point which lies on all of them