Read the following axioms
(i) Things which are equal to the same thing are equal to one another
(ii) If equals are added to equals, the wholes are equal
(iii) Things which are double of the same thing are equal to one another
Check whether the given system of axioms is consistent or inconsistent
Solution:
Given the statements are
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.
(iii) Things which are double of the same thing are equal to one another.
We have to determine if the system of axioms is consistent or inconsistent.
A system of axioms is called consistent , if there is no statement which can be deduced from these axioms such that it contradicts any axiom.
According to Euclid’s first axiom,
The things which are equal to the same thing are equal to one another.
According to Euclid’s second axiom,
If equals are added to the equals, the wholes are equal.
According to Euclid’s sixth axiom,
Things which are double of the same thing are equal to one another.
We observe that the given statements are Euclid’s axioms.
We cannot deduce any statement from these axioms which contradicts any axiom.
Therefore, the given statements are consistent.
✦ Try This: In the adjacent figure, if ∠1 = ∠3, ∠2 = ∠4 and ∠3 = ∠4, write the relation between ∠1 and ∠2 using an Euclid’s postulate
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 5
NCERT Exemplar Class 9 Maths Exercise 5.4 Problem 5
Read the following axioms: (i) Things which are equal to the same thing are equal to one another, (ii) If equals are added to equals, the wholes are equal, (iii) Things which are double of the same thing are equal to one another. Check whether the given system of axioms is consistent or inconsistent
Summary:
From the following axioms: (i) Things which are equal to the same thing are equal to one another, (ii) If equals are added to equals, the wholes are equal, (iii) Things which are double of the same thing are equal to one another. The given system of axioms is consistent as they belong to Euclid’s axioms
☛ Related Questions:
- Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Ch . . . .
- Read the following statements which are taken as axioms : (i) If a transversal intersects two parall . . . .
- Read the following two statements which are taken as axioms: (i) If two lines intersect each other, . . . .