Reasoning and Two-Column Proofs

A chain of statements where every step is justified by a definition, postulate, theorem, or property — never by how the picture looks.

A postulate is an accepted starting assumption (no proof needed); a theorem is a statement that has already been proven from postulates and definitions.

Reasoning from specific observations to a general conjecture (e.g., 3, 6, 9, 12 → next is 15). It can be wrong and is disproven by a single counterexample.

Reasoning from accepted general rules to a guaranteed specific conclusion. Proofs are deductive.

If p → q is true and p is true, then q must be true.

If p → q and q → r are true, then p → r is true (chaining conditionals).

Converse: q → p; Inverse: ~p → ~q; Contrapositive: ~q → ~p.

Its contrapositive — a conditional and its contrapositive always have the same truth value. The converse is a separate claim.

'p if and only if q' — used when both a statement and its converse are true. Good definitions are biconditional.

If B is between A and C, then AB + BC = AC. If D is interior to ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

Left column = Statements, right column = Reasons. Line 1 is the Given; the last line is what you were asked to prove; every statement needs a valid reason.

It lets you state that a shared side or angle is congruent to itself (AB = AB, ∠A ≅ ∠A).