These comprehensive RBSE Class 9 Maths Notes Chapter 5 Introduction to Euclid’s Geometry will give a brief overview of all the concepts.
RBSE Class 9 Maths Chapter 5 Notes Introduction to Euclid’s Geometry
1. Though Euclid defined a point, a fine, and a plane, the definitions are not accepted by mathematicians. Therefore, these terms are now taken as undefined.
2. Euclid’s definitions :
(i) A point is that which has no part, i.e. a point is an exact location.
(ii) A line is breadthless length.
(iii) The ends of a line are points.
(iv) A straight line is a line which lies evenly with the points on itself.
(v) A surface is that which has length and breadth only.
(vi) The edges of a surface are lines.
(vii) A plane surface is a surface which lies evenly with the straight lines oil itself.
Point: A small dot marked by a sharp pencil on a sheet of paper or a prick made by a fine needle on a paper are examples of a point. A point determines a location in space. It has no length, breadth or thickness.
Line : A line has length only. It has no breadth or thickness. The basic concept of a line is its straightness and it extends infinitely in both directions.
The two arrowheads in the opposite directions indicate that the length of the line is unlimited i.e. it has no definite length. A line has no end points and it consists of an infinite (uncountable) number of points.
Plane : A plane has length and breadth. It has fio thickness. The basic concept of a plane is its flatness and it extends indefinitely in all directions. The length and breadth of a plane are unlimited i.e. a plane has no definite length and no definite breadth. For example : A surface of a smooth blackboard, the table top.
3. Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.
4. Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.
5. Some of Euclid’s axioms were :
(a) Things which are equal to the same things are equal to one another.
(b) If equals are added to equals, the wholes are equal.
(c) If equals are substracted from equals, the remainders are equal.
(d) Things which coincide with one another are equal to one another.
(e) The whole is greater than the part.
(f) Things which are double of the same things are equal to one another.
(g) Things which are halves of the same things are equal to one another.
6. Euclid’s postulates were :
Postulate 1: A straight line may be drawn from any one point to any other point.
Postulate 2: A terminated line can be produced indefinitely.
Postulate 3: A circle can be drawn with any centre and any radius.
Postulate 4 : All right angles are equal to one another.
Postulate 5 : If a straignt line 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.
7. Two equivalent versions of Euclid’s fifth postulate are :
(i) ‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to
(ii) Two distinct intersecting lines cannot be parallel to the same line.
8. All the attempts to prove Euclid’s fifth postulate lasing the first 4 postulates failed. But they led to the discovery of several other geometries, called non-Eudidean geometries.