Introduction to Proofs

Geometry · Axiom Academy

Building logical arguments in geometry — from what you're given to what you've shown. In geometry, we don't just believe — we prove A geometric proof is a logical argument that uses definitions, postulates, and previously proven theorems to show that a statement must be true. We don't accept a claim because a picture looks right — we force it from facts we already trust. Here's the whole idea on one triangle: from a single given, a conclusion we never assumed. Given a triangle with two equal sides ( ), watch its base angles turn out equal too. The figure folds along its axis of symmetry: the equal sides swap places, so the angles they sit against must match. That forced match — not the drawing looking symmetric — is the proof. One given fact ( ) forced a new one ( ). That forcing — fact compelling fact — is exactly what a proof does. Given, statements, reasons, conclusion Strip a proof to its frame and it always has the same parts: the given , a column of statements each paired with the reason that justifies it, and the conclusion . Watch the same triangle proof get written this way — and as each row lands, the part of the figure it's talking about lights up. Every statement on the left earns a reason on the right. No naked claims — that pairing is what makes it a proof. A first proof: vertical angles are equal

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