Two-Column Proofs: How to Write Statements and Reasons

A two-column proof is a logical argument arranged in a table: the left column states each fact, and the right column explains why that fact is true. Start from the givens and finish with the statement you were asked to prove. A reason can be a definition, a property, a postulate, or a theorem you are allowed to use.

How to write a two-column proof

1. Separate the given from the goal. Write down what you know and what you must show. Mark only the information stated or indicated by the diagram; a picture that looks symmetrical is not a reason.
2. Plan a short route. Ask which definition or theorem would establish the goal, then look for the facts it needs. Planning backward is useful, but write the finished proof forward from known facts.
3. Pair each statement with its reason. When a step uses earlier rows, identify them. A step may depend on several earlier facts, not just the row directly above it.
4. Reach the exact conclusion. If the goal is angle congruence, do not stop at equal angle measures. Use the definition of congruent angles for the final step.

Notation: ∠A names an angle; m∠A is its measure. Write m∠A = m∠B for equal measures and ∠A ≅ ∠B for congruent angles. AB means the length of segment AB in the examples below.

Common reasons and when to use them

• Given: information stated in the problem.
• Definition: translate a geometric term into usable facts. Supplementary angles have measures that add to 180°; complementary angles have measures that add to 90°.
• Substitution: replace a quantity with an equal quantity.
• Subtraction property of equality: subtract the same quantity from both sides of an equation.
• Symmetric property of equality: if a = b, then b = a.
• Definition of congruent angles: angles with equal measures are congruent.

Use the names and permitted results from your class. “It looks equal” and “algebra” do not explain which fact or operation justifies a step.

Worked example: angles supplementary to the same angle

Given: ∠A and ∠C are supplementary; ∠B and ∠C are supplementary.
Prove: ∠A ≅ ∠B.

The shared angle measure cancels in row 5. No particular size for ∠C is needed. Row 6 then connects the algebraic equality to the geometric conclusion.

Practice two-column proofs

Try these on paper before opening the solutions. Draw two columns labeled Statement and Reason; number the rows so you can refer to earlier facts.

1. Given AB = CD, prove CD = AB.
2. Given ∠X and ∠Y are complementary, and ∠Y and ∠Z are complementary, prove ∠X ≅ ∠Z.
3. A student says, “The angles look equal in the drawing, so they are congruent.” Explain what is missing.

Check your proof before handing it in

Read each row as “This is true because…” Check that its reason uses facts already established, that every equation preserves equality, and that the last row matches the goal. Do not use the result you are trying to prove as its own justification. More than one valid proof may be possible; a longer proof is not automatically a better proof.

If angle vocabulary is the sticking point, review complementary and supplementary angles. For the next geometry topic, use the Geometry course topic index.