Converse of a Theorem

a theorem obtained by reversing the roles of premise and conclusion of the initial (direct) theorem. Any theorem is the converse of its converse. Thus, a theorem and its converse are converses of each other. For example, the theorems “If two angles of a triangle are equal, then their bisectors are equal” and “If two bisectors of a triangle are equal, then the corresponding angles are equal” are converses of one another.

In general, the validity of a theorem does not imply the validity of its converse. For example, the theorem “If a number is divisible by 6, then it is divisible by 3” is true, but its converse, “If a number is divisible by 3, then it is divisible by 6,” is false.

The means used to prove a theorem may be insufficient to prove its valid converse. For example, in Euclidean geometry both the theorem “Two lines in a plane that have a common perpendicular do not intersect” and its converse, “Two noninter-secting lines in a plane have a common perpendicular,” are true. However, the proof of the second (converse) theorem is based on Euclid’s parallel postulate, whereas this postulate is unnecessary to prove the first theorem. In Lobachevskian geometry the first theorem is true and the second is false.

The converse of a theorem is equivalent to the opposite of the direct theorem, that is, the theorem in which the premise and conclusion of the direct theorem are replaced by their negations. Therefore, the direct theorem is equivalent to the opposite of its converse, that is, the theorem that asserts that if the conclusion of the direct theorem is false, then so is its premise. The well-known method of indirect proof (reductio ad absurdum) is merely the replacement of the proof of a theorem by that of the opposite of its converse. If a theorem and its converse are valid, then the premise of either theorem is not only sufficient but also necessary for the validity of the conclusion. (SeeNECESSARY AND SUFFICIENT CONDITIONS.)