Conditional Statements & Logic

Geometry runs on if-then statements: "If it is a square, then it is a rectangle." The if part is the hypothesis, the then part is the conclusion.

Three statements are built from any conditional:

• Converse: swap them. "If it is a rectangle, then it is a square."
• Inverse: negate both. "If it is not a square, then it is not a rectangle."
• Contrapositive: swap *and* negate. "If it is not a rectangle, then it is not a square."

The important fact: a statement and its contrapositive always have the same truth value. The example above is true, and so is its contrapositive. But its converse is false — plenty of rectangles are not squares.

When a statement and its converse are both true, you get a biconditional, written "if and only if". Every definition in geometry is a biconditional.

For a conditional p → q: the converse is q → p, the inverse is ¬p → ¬q, and the contrapositive is ¬q → ¬p.

A conditional is logically equivalent to its contrapositive, and the converse is equivalent to the inverse. A conditional and its converse are independent — either can hold without the other, which is precisely why every geometry theorem's converse must be proved separately rather than assumed.

When both p → q and q → p hold, the biconditional p ↔ q is true. Definitions are always biconditional even when written as one-directional sentences: "a midpoint divides a segment into two congruent segments" is understood to work both ways, which is what allows a definition to be used in either direction inside a proof.

The two reasoning modes used in the course: deductive reasoning applies accepted facts to reach a guaranteed conclusion, while inductive reasoning generalises from observed patterns and only produces a conjecture. A single counterexample disproves a conjecture, whereas no number of confirming examples proves one — the reason a proof is required at all.