DWG NO. 02.4 — Lesson 4 of 5

Introduction to Two-Column Proofs

Unit 2: Reasoning and Proof · ~15–30 min

Objective

Write a two-column proof that supports each geometric statement with a valid reason — given information, a definition, a postulate, or a theorem.

Anatomy of a two-column proof

A two-column proof starts with a Given (what you know) and a Prove (what you're trying to show). Below that, a left column lists each logical statement in order, and a right column gives the reason that justifies it. Every single statement needs a reason — there's no such thing as an unsupported step.

Valid reasons

Type	Example
Given	Information stated directly in the problem
Definition	Definition of midpoint, definition of congruent angles, …
Postulate	Segment Addition Postulate, Linear Pair Postulate, …
Theorem	A previously proven result, cited by name
Algebra	Addition, Subtraction, Substitution, and other properties of equality (Lesson 2.5)

∠1 and ∠3 are vertical angles formed by two intersecting lines — the pair proved congruent in Worked Example 1.

Worked Example 1 · Vertical angles are congruent

ProblemGiven: ∠1 and ∠3 are vertical angles, formed by two intersecting lines as in the diagram above (so ∠1/∠2 and ∠2/∠3 are each linear pairs). Prove: ∠1 ≅ ∠3.

1.Statement: m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180°. — Reason: Linear Pair Postulate.

2.Statement: m∠1 + m∠2 = m∠2 + m∠3. — Reason: Transitive Property of Equality (both equal 180°).

3.Statement: m∠1 = m∠3. — Reason: Subtraction Property of Equality (subtract m∠2 from both sides).

4.Statement: ∠1 ≅ ∠3. — Reason: Definition of congruent angles.

∠1 ≅ ∠3 — proved

Worked Example 2 · Midpoint proof

ProblemGiven: M is the midpoint of segment AB. Prove: AM = MB.

1.Statement: M is the midpoint of segment AB. — Reason: Given.

2.Statement: AM = MB. — Reason: Definition of midpoint (a midpoint splits a segment into two congruent, equal-length parts).

AM = MB — proved directly from the definition

Guided practice

1.What reason justifies the statement “AB = AB”?
Show answer
Reflexive Property of Equality — any quantity is equal to itself.
2.In a two-column proof, can a later statement use an earlier statement in the same proof as its reason?
Show answer
Yes — once a statement has been proven earlier in the proof, it becomes a fact you can cite as a reason for a later step.
3.Given the reason “Definition of segment bisector,” what kind of statement would it justify?
Show answer
A statement saying two segments created by the bisector are congruent (or equal in length), e.g. “segment AC ≅ segment CB.”
4.Write a short two-column proof: Given ∠A ≅ ∠B and ∠B ≅ ∠C, prove ∠A ≅ ∠C.
Show answer
1. ∠A ≅ ∠B and ∠B ≅ ∠C — Given. 2. ∠A ≅ ∠C — Transitive Property of Congruence.
5.Why must every statement in a two-column proof have a reason, even ones that seem obvious?
Show answer
A proof has to be built entirely on facts already accepted as true — definitions, postulates, theorems, or given information — so that no step relies on something “looking” true. Skipping a reason breaks the chain of logical certainty.

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