Q3 of 48 Page 52

Read the following statements which are taken as axioms:

(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.


(ii) If a transversal intersects two parallel lines, then alternate interior angles are equal.


Is this system of axioms consistent? Justify your answer.

A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axioms.

We know that, if a transversal intersects two parallel line, then each pair of corresponding angles are equal, which is a theorem. So, statement (i) is false and not an axiom.


Also, we know that, if a transversal intersects two parallel line, then each pair of alternate interior angles are equal. It is also a theorem. So, statement(ii) is true and an axiom.


Thus, in given statements, first is false and second is an axiom.


Hence, given of axioms is not consistent.


More from this chapter

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1

Read the following statement:

An equilateral triangle is a polygon made up of three-line segments out of which two-line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?

 2

Study the following statement:

“Two intersecting lines cannot be perpendicular to the same line”.


Check whether it is an equivalent version to the Euclid’s fifth postulate.


[Hint: Identify the two intersecting lines l and m and the line n in the above statement.]

 4

Read the following two statements which are taken as axioms:

(i) If two lines intersect each other, then the vertically opposite angles are not equal.


(ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°.


Is this system of axioms consistent? Justify your answer.

 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.