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.

Common reasons for two-column proofs

Almost every proof in a first geometry course uses reasons from this list. Keep it next to you while you practice.

Full worked proof: triangles that meet at a midpoint

Given: Segments AB and CD cross at point M. M is the midpoint of AB and the midpoint of CD.

Prove: △AMC ≅ △BMD

Notice that the angle in step 4 sits between the two sides from steps 2 and 3. That is what makes SAS the right reason. Once the triangles are congruent, you could add a step 6, AC ≅ BD, with the reason CPCTC.

Frequently Asked Questions

What is a two-column proof in geometry?

A two-column proof is a geometry argument written as a table. The left column lists numbered statements, and the right column gives the reason each statement is true, such as Given, a definition, a postulate or a theorem. The first rows usually restate the given facts, and the last row is exactly the statement you were asked to prove.

How do you write two-column proofs in geometry?

Write the given facts and the goal first, then mark the diagram with only what is stated. Plan backward: ask which theorem would prove the goal and what facts it needs. Then write the proof forward, one statement per row, each with its reason. When a step depends on earlier rows, cite them, and make sure your final row matches the goal word for word.

How do you do two-column proofs in geometry with angles?

Angle proofs usually move between measures and congruence. Suppose ∠1 and ∠2 are supplementary and ∠3 and ∠2 are supplementary. Write m∠1 + m∠2 = 180° and m∠3 + m∠2 = 180° by the definition of supplementary angles, then m∠1 = m∠3 by substitution and subtraction. Finish with ∠1 ≅ ∠3 by the definition of congruent angles, or cite the Congruent Supplements Theorem.

How do you do two-column proofs in geometry with triangles?

First collect three pairs of matching parts, then name a congruence rule such as SSS, SAS, ASA, AAS or HL. For example, if M is the midpoint of both AD and BC, then AM ≅ DM and BM ≅ CM by the definition of midpoint, and ∠AMB ≅ ∠DMC as vertical angles, so the triangles are congruent by SAS. After that, CPCTC proves any other matching parts congruent.

Why is a two-column proof a good format choice?

The two-column format makes every step and its justification easy to see. Because each statement must sit next to a reason, it is hard to skip a step or rely on how the diagram looks. Teachers can quickly spot where an argument breaks, and students learn to separate what they know from why they know it, a habit that carries into later proofs.

How do you write statements and reasons in geometry?

A statement is a specific fact about the figure, written with correct notation, such as AB = CD or ∠A ≅ ∠B. Its reason names the rule that makes it true, such as Given, Definition of midpoint, Segment Addition Postulate or Vertical Angles Theorem. Keep reasons general rules, not descriptions of the picture, and avoid saying something is true because it looks that way.