Circular Reasoning in Proofs MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

Circular reasoning is a logical fallacy.
It occurs in Geometry when two statements depend upon each other to be true.

Statement 1 is used to prove Statement 2,
but
Statement 2 is used to prove Statement 1.

In Geometry, there is a hierarchy (a specific order) as to when theorems are proven to be true in relation to one another. A theorem needs to be proven true "before" it can be used to prove another theorem true. This helps avoid circular reasoning.

When proving theorems true in Geometry, you must be careful not to use circular reasoning.

Consider these two theorems:

1: In a plane, if a line is perpendicular to one of two parallel lines, it is also perpendicular to the other line.

2: If two lines are parallel, then the corresponding angles are congruent.

Circular Reasoning:

Theorem 1 is used to prove Theorem 2
(says theorem 1 must be above theorem 2 in the hierarchy)
but then
Theorem 2 is used to prove Theorem 1.
(says theorem 2 must be above theorem 1 in the hierarchy?????)

Under these conditions, Theorem 2 would not be considered a valid step in proving Theorem 1.

FYI: This web site will be using Theorem 1 to prove Theorem 2.
But we will NOT be using Theorem 2 to prove Theorem 1.

Our proof of Theorem 1 will not involve the use of corresponding angles.
Theorem 1 is proven under Perpendicular Lines using an Indirect Proof and the Parallel Postulate.
This more generic proof of Theorem 1 will allow for a wider application of that theorem
in other proofs, and will maintain Theorem 1 above Theorem 2 in the hierarchy.