Two-Column Proofs

Geometry · Axiom Academy

The most structured format for geometric proofs — every statement, paired with the reason it's true. 1. Statements and Reasons, Side by Side The whole idea of a two-column proof is a promise: nothing is asserted without justification . Each row pairs a statement you are claiming with the reason — a given, definition, postulate, theorem, or property — that makes it true. Every two-column proof follows the same shape: start from what you are given , add one justified step at a time, and finish with the statement you set out to prove . Watch one assemble on a short example — M is the midpoint of , so . 3. Example: Vertical Angles Are Congruent Two lines cross at a point, forming four angles. We prove that a pair of opposite ( vertical ) angles must be congruent. Play the proof and watch each statement light up the part of the figure it talks about. Given: Lines AB and CD intersect at point E We use the Linear Pair Postulate twice (steps 3 and 6) to build two equations that share 2 . Setting them equal (step 8) and subtracting m 2 from both sides (step 9) eliminates the shared angle and leaves exactly m 1 = m 3 . A statement is only valid if its reason comes from an approved list — a given, a definition, a property of equality or congruence, or a proven theorem. Watch a claim get tested against three candidate reasons: only the one that truly justifies it is allowed to dock. Here is the full toolkit you draw from. 5. Example: Using Segment Addition

This is the written version of the interactive lesson above. See the full Geometry course.