Paragraph Proofs

Geometry · Axiom Academy

Writing the same logical argument as flowing, connected sentences 1. A Proof That Reads Like Writing A two-column proof stacks each statement next to its reason . A paragraph proof carries that exact same chain of reasoning, but as connected sentences : each statement is written out, its reason folded in, and the steps joined by linking words. Here is the simplest case — a one-line midpoint argument. Play it to watch the reasoning travel across the figure, one beat at a time. The argument, written as a paragraph ▶︎ Play the animation to reveal the written proof The two formats trade off structure against flow. A two-column proof is rigidly organized and easy to check line by line, which makes it great for learning. A paragraph proof is more concise and reads like natural mathematical writing, which is why it dominates in higher math. The animation traces the same two-line proof on one right angle — first as two columns, then flowing as a single sentence. Statements and reasons separated Take , a right angle. Two-column: (1) is a right angle [Given]; (2) [Reflexive Property of Congruence]. Paragraph: "We are given that is a right angle. By the Reflexive Property of Congruence, any angle is congruent to itself. Therefore, . ∎ " ▶︎ Play the animation to reveal both formats 3. The Structure: Given → Steps → Conclusion

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