Question

Read the following statements which are taken as axioms:

1. If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
2. If a transversal intersect two parallel lines, then alternate interior angles are equal. 

Is this system of axioms consistent? Justify your answer.

Answer 1

A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axiom. We know that, if a transversal intersects two parallel lines, 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 lines, 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 system of axioms is not consistent.