Logic in geometry: conditionals, converses, contrapositives, and counterexamples

Geometry is built on logic. Undefined terms, like point, line, and plane, are described rather than defined; definitions give precise meanings to other terms; postulates are statements accepted without proof; conjectures are unproven guesses; and theorems are statements that have been proved. A conditional statement has the form "If p, then q." Its converse is "If q, then p," its inverse is "If not p, then not q," and its contrapositive is "If not q, then not p." A conditional and its contrapositive are always both true or both false, but the converse and inverse can differ from the original. For example, "If an angle measures 30 degrees, then it is acute" is true, but its converse, "If an angle is acute, then it measures 30 degrees," is false. A single counterexample, like a 40-degree angle, is enough to prove a statement false. When a conditional and its converse are both true, they combine into a biconditional, "p if and only if q," which is how good definitions work.

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