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

Solution:

Given, the statements are

(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

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.

The alternate interior angle theorem states that if a transversal intersects two parallel lines, then corresponding angles are equal.

From the figure,

∠1 = ∠5,

∠2 = ∠6

∠3 = ∠7

∠4 = ∠8

Therefore, the first statement is false and not an axiom.

The alternate interior angle theorem states that if a transversal intersects two parallel lines, then the alternate interior angles are equal.

Therefore, the second statement is true and an axiom.

✦ Try This: List Euclid’s five postulates.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 5

NCERT Exemplar Class 9 Maths Exercise 5.4 Problem 3

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

Summary:

From the following statements (i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal, is not consistent and not an axiom, (ii) If a transversal intersects two parallel lines, then alternate interior angles are equal is consistent and an axiom

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