Paragraph Proof — Definition, Formula & Examples
A paragraph proof is a mathematical proof written as a continuous block of sentences rather than in a table or diagram. Each sentence states a claim and its justification, building logically from the given information to the conclusion.
A paragraph proof is a deductive argument in which statements and their corresponding reasons are expressed in narrative prose, maintaining a strict logical sequence from hypotheses to the desired conclusion, with every assertion supported by a definition, postulate, or previously proven theorem.
How It Works
Start by stating what is given and what you need to prove. Then write a series of connected sentences, each making one logical claim and immediately citing the reason (a definition, postulate, or theorem) that supports it. Every new statement should follow from earlier ones, creating a chain of reasoning that ends at the conclusion. The key difference from a two-column proof is format, not rigor — you still must justify every step.
Example
Problem: Given: ∠A and ∠B are complementary, and m∠A = 35°. Prove: m∠B = 55°.
Write the proof: We are given that ∠A and ∠B are complementary and that m∠A = 35°. By the definition of complementary angles, m∠A + m∠B = 90°. Substituting the given measure, 35° + m∠B = 90°. By the Subtraction Property of Equality, m∠B = 90° − 35° = 55°. Therefore, m∠B = 55°.
Answer: The paragraph proof shows that m∠B = 55° using the definition of complementary angles and the Subtraction Property of Equality.
Why It Matters
Paragraph proofs appear frequently on geometry exams and standardized tests, often alongside two-column and flowchart proofs. Learning to write them strengthens your ability to communicate mathematical reasoning clearly — a skill used in college math courses and any career that requires building logical arguments from evidence.
Common Mistakes
Mistake: Writing claims without stating the reason (theorem, postulate, or definition) that justifies each one.
Correction: Every statement in a paragraph proof must be paired with its justification. After each claim, include a phrase like 'by the definition of…' or 'by the … theorem' so the reader can verify your logic.