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.

Question

Philoid · Accepted 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.