Class 9 Maths Chapter 5
Class 9 maths chapter 5 (Introduction to Euclid’s Geometry). In this chapter we learnt the postulate and axioms of Euclid’s.Axioms or postulate are the assumptions which are obvious universal truths.They are not proved. But the theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning.
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Exercise 5.1 and download the PDF of Exercise 5.1
Exercise 5.2 and download the PDF of Exercise 5.2
Class 9 Maths Chapter 5 Euclid's Axioms:
Some of the Euclid’s axioms are given below:
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are equal.
(3) If equals are subtracted from equals, the remainders are equal.
(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.
Exercise 5.1 Class 9 NCERT Solutions:
Click here for the solution of Exercise 5.1, Exercise 5.2 and download the PDF of Exercise 5.1, Exercise 5.2
Exercise 5.2 Class 9 NCERT Solutions:
Click here for the solution of Exercise 5.1, Exercise 5.2 and download the PDF of Exercise 5.1, Exercise 5.2
Class 9 Maths Chapter 5 Postulates:
Euclid’s postulates were:
(1) A straight line may be drawn from any one point to any other point.
(2) A terminated line can be produced indefinitely.
(3) A Circle can be drawn with any centre 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 two straight lines, if produced indefinitely , meet on that side on which the sum of angles is less than two right angles.
Two equivalent versions of Euclid’s fifth postulate in class 9 maths chapter 5 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 l.
(ii) two distinct intersecting lines cannot be parallel to the same line.
Euclid's definitions:
Euclid summarised some statements as definition. He began his exposition by listing 23 definition in Book 1 of the ‘Element’. A few of them are given below:
1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are 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 breadth only.
6. The edge of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.
If you care fully study these Euclid’s definitions in class 9 maths chapter 5 , you find that some of the terms like part, breadth, length, evenly, etc.need to be further explained clearly. For example, consider his definition of a point . In this definition, ‘a part’ needs to be defined. let you define ‘a part’ to be that which occupies ‘area’ , again ‘an area’ needs to be defined . So, to define one thing, you need to define many other things, and you may get a long chain of definition.
For such reasons, mathematicians agree to leave some geometric terms undefined. A similar problem arises in definition 2 above, since it refers to breadth and length, neither of which has been defined.
Therefore a few terms are kept undefined while developing any course of study. So, in geometry, we take a point, a line and a plane (in Euclid’s words a plane surface ) as undefined terms. The only thing is that we can represent them intuitively, or explain them with the help of ‘physical models’ .