Writing Two-Column Proofs

A two-column proof lists statements on the left and the reason for each on the right. The first statements are the givens; the last is what you were asked to prove.

Every reason must be one of four things: a given, a definition, a postulate, or a theorem already proved. "It looks that way" is never a reason.

How to get started when you are stuck:
1. Mark the diagram with everything you are given.
2. Write the goal at the bottom and ask what would immediately produce it.
3. Work backwards from the goal and forwards from the givens until they meet.

Three reasons carry a surprising amount of the work: the Reflexive Property (a shared side is congruent to itself), vertical angles, and the definition of a midpoint or bisector.

Diagrams may be trusted for which points lie between others, but never for lengths or angle sizes.

A proof is a finite chain of statements, each justified, leading from hypothesis to conclusion. The two-column format is a bookkeeping device; paragraph and flowchart proofs carry identical logical content.

The properties of equality and congruence supply many reasons: reflexive (a = a), symmetric, transitive, and the substitution property. The reflexive property is the one students overlook, and it is exactly what licenses using a shared side in a triangle congruence proof.

Deciding what may be assumed from a diagram is a real skill. Collinearity, betweenness and intersection may be read off. Congruence, parallelism, perpendicularity and right angles may not, unless marked or given.

Strategically, most proofs are found by working backwards. Identify the last step — the theorem whose conclusion is exactly the goal — then treat that theorem's hypotheses as the new goals and recurse. Proofs of segment or angle congruence typically route through triangle congruence and CPCTC, which is why that pair dominates the middle of the course.