Geometric Proofs Summary

Geometry · Axiom Academy

SUMMARY Unit 10 Summary: Geometric Proofs The reasoning toolkit — how a chain of justified statements turns a picture into a guarantee. Inductive reasoning spots a pattern and offers a probable conjecture; deductive reasoning chains accepted facts to a certain conclusion. Proofs are deductive. The two engines of deduction: the Law of Detachment (if "if p then q " is true and p is true, conclude q ) and the Law of Syllogism (chain "if p then q " with "if q then r " to get "if p then r "). The same proof can be written three ways — a two-column proof , a paragraph proof , or a flowchart proof . They differ only in format, never in logic. Every statement needs a reason. The legal reasons are exactly four kinds: definitions, postulates, theorems, and the properties of equality and congruence. The workhorse properties: reflexive ( a = a ), symmetric ( ), transitive ( ), and substitution . Core Concept Inductive vs. Deductive Reasoning Inductive reasoning generalizes from specific observations — useful for forming a conjecture, but never a proof. Deductive reasoning moves from general truths to a specific conclusion that is guaranteed. Inductive: " so the next is 10 " — likely, not certain. Watch out for: one counterexample destroys an inductive conjecture; a valid deduction has none. Core Concept Law of Detachment

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