Geometry Proofs

A PROOF is a logical argument that shows WHY something in geometry MUST be true. Proofs are how mathematicians make certain claims. If you can prove a geometry theorem, you can structure any argument in life.

Every proof has:\n\n- **GIVEN** — what you're told is true\n- **PROVE** — what you're trying to show\n- **STATEMENTS** — numbered logical steps\n- **REASONS** — justification for each step\n\nFormat as a two-column table with statements on the left, reasons on the right.

- Given — stated in the problem\n- Definition of [midpoint, parallel, etc.]\n- Reflexive property — anything equals itself\n- Vertical angles are congruent\n- Corresponding angles congruent (parallel lines)\n- SSS, SAS, ASA, AAS, HL — triangle congruence\n- CPCTC — corresponding parts of congruent triangles are congruent\n- Substitution, Transitive, Addition properties

Given: M is midpoint of AB. Prove: AM = MB.\n\n1. M is midpoint of AB — Given\n2. AM ≅ MB — Definition of midpoint\n3. AM = MB — Definition of congruent segments\n\nShort, but a complete logical chain.

Two triangles are CONGRUENT if:\n\n- SSS — all 3 sides\n- SAS — 2 sides + included angle\n- ASA — 2 angles + included side\n- AAS — 2 angles + non-included side\n- HL — hypotenuse + leg (right triangles)\n\nThese appear in 80% of HS geometry proofs.

- Draw and mark GIVENS on the figure\n- Start from the givens\n- Work backwards from what you need to prove\n- Use congruent triangles to unlock CPCTC\n- Every statement needs a reason\n- Be precise: ≅ means congruent, = means equal

- Two-column (common in HS)\n- Paragraph (connected sentences)\n- Flowchart (visual)\n- Indirect / proof by contradiction\n\nAll prove the same thing. Choose by style or assignment.