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  • 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 intersect two parallel lines, then alternate interior angles are equa

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 intersects two parallel lines, then alternate interior angles are equal.

Is this system of axioms consistent? Justify your answer.

Answers (1)

Given the system of axioms is not consistent.

Solution:

Axiom: A statement or proposition which is regarded as being self-obviously evident, established or accepted. Hence, it is accepted without controversy or question.

Statement (i) is False.

We know that if a transversal intersects two parallel lines, then each pair of corresponding angles is equal. It is a theorem.

Theorem: A statement or proposition not plainly obvious but rather demonstrated by a chain of reasons; a fact set up by methods for acknowledged certainties

Statement (ii) is true.

If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. It is also a theorem.

In the given figure, \angle3 = \angle6, \angle4 = \angle5  (alternate interior)

\angle1 = \angle5, \angle2 = \angle6,   (corresponding angles)

\angle3 = \angle7, \angle4 = \angle8     (corresponding angles)

Hence statement (i) is not an axiom.

Statement (ii) is true and an axiom

Hence, the given system of axioms is not consistent.

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