A two-column proof is a way of writing a proof as a table: numbered statements in the left column and the reason for each one in the right column. It begins with the given facts and ends with the statement to be proved. Every reason must be a given, a definition, a postulate, a property or a proved theorem.
In Spanish
demostración a dos columnas
Also called
statement-reason proof, T-proof
Grade level
Geometry
GeometryCCSS HSG-CO.C.9CCSS HSG-CO.C.10Praxis
Two-Column Proof in a picture
A two-column proof is a numbered checklist: each step on the left, its justification on the right, ending with the Prove statement.
Formal definition and notation
A two-column proof of a conditional P ⇒ Q lists statements S1, S2, …, Sn beside reasons R1, R2, …, Rn. S1 is usually the hypothesis P (reason: given), Sn is the conclusion Q, and each Rk names the definition, postulate, property of equality or congruence, or earlier theorem that justifies Sk from statements above it.
Symbol
Read it as
Meaning
Given
given
information stated in the problem; it needs no other justification
Prove
prove
the statement that must appear as the last row
≅
is congruent to
same size and shape; common in geometry statements
4. ∠ 2 = 115°. Reason: Subtraction Property of Equality.
Answer:∠ 2 = 115°, proved in 4 steps.
Example 3: Fill in the missing reason
Test-style
Given: M is the midpoint of AB, and AM = 4x − 1, MB = 2x + 9. Statement 2 reads AM = MB. What is its reason, and what is AB?
A midpoint splits a segment into two equal parts, so the reason for AM = MB is the Definition of midpoint.
Substitution: 4x − 1 = 2x + 9, so 2x = 10 and x = 5.
AM = 4(5) − 1 = 19, so AB = AM + MB = 19 + 19 = 38 (Segment Addition Postulate).
Answer: The reason is the Definition of midpoint, and AB = 38.
Common mistakes
✗ MistakeWriting a statement with no reason, or the reason ‘because it looks equal’.
✓ InsteadGive every row a reason from the accepted list, such as Given or Definition of angle bisector.
The reasons column is what makes the argument checkable; appearance is never a reason.
✗ MistakeCiting the Division Property for 2x + 10 = 26 → 2x = 16.
✓ InsteadSubtracting 10 from both sides uses the Subtraction Property of Equality.
Name the operation actually performed on both sides of the equation.
✗ MistakeStarting with the Prove statement as step 1 (reason: Given).
✓ InsteadStart with the given facts and reach the Prove statement in the last row.
Assuming the conclusion makes the proof circular.
Not to be confused with
Two-Column Proofvs.Paragraph proof
A paragraph proof writes the same statements and reasons in full sentences instead of a table. The logic is identical; only the format changes.
Two-Column Proofvs.Flowchart proof
A flowchart proof puts each statement in a box with its reason underneath and draws arrows to show which steps lead to which.
Quick check: 3 questions
Question 1 of 3
In a two-column proof, the step x + 4 = 10 → x = 6 is justified by:
Show answer and explanation
Answer: B. Subtracting 4 from both sides is the Subtraction Property of Equality. Addition would undo a subtraction, the Distributive Property removes parentheses, and x = 6 is not given.
Question 2 of 3
Which reason justifies PQ ≅ PQ when two triangles share side PQ?
Show answer and explanation
Answer: C. Anything is congruent to itself: the Reflexive Property. Symmetric flips an equation (if a = b, then b = a), Transitive chains two, and a midpoint is unrelated.
Question 3 of 3
What must the last statement of a two-column proof be?
Show answer and explanation
Answer: B. The proof is complete only when it reaches the Prove statement. The given comes first, and a definition is a reason, not the goal.
Two-Column Proof: frequently asked questions
How do you write a two-column proof?
Draw two columns, Statements and Reasons. Write the given information first, with the reason Given. Then add one small step per row, each following from rows above, and name its justification: a definition, postulate, property or theorem. Keep going until the last statement is exactly the Prove statement. Number the rows so later reasons can refer to them.
What reasons can you use in a two-column proof?
Given information, definitions (such as midpoint, right angle or angle bisector), postulates (such as Segment Addition), algebra properties (Addition, Subtraction, Multiplication, Division, Substitution, Distributive, Reflexive, Symmetric, Transitive), and theorems already proved (such as the Vertical Angles Theorem). You cannot use measurements from the picture or the statement you are proving.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSG-CO.C.9, CCSS HSG-CO.C.10
Cite this entry
Effortless Math Team. “Two-Column Proof.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/two-column-proof/
Flashcard
TermTwo-Column Proofstatements | reasons
DefinitionA proof written as a table: each statement on the left, its reason on the right. Example: 2x = 16, so x = 8 (Division Property).
ProofA proof is a step-by-step logical argument that shows a math statement is true in every case it covers. It starts from given facts and uses definitions, postulates and statements already proved, with a reason for each step. A statement that has been proved is called a theorem.Geometry · Geometry, Algebra 1TheoremA theorem is a math statement that has been proved true using definitions, postulates and statements already proved. Once proved, it is true in every case it covers, so you can use it to solve problems. The Pythagorean theorem and the Vertical Angles Theorem are famous examples.Geometry · Grade 8, GeometryMidpointA midpoint is the point exactly halfway between the two endpoints of a line segment, so it splits the segment into two equal parts. On the coordinate plane, find it by averaging the coordinates: the midpoint of (x1, y1) and (x2, y2) is (x1 + x22, y1 + y22).Geometry · Grade 8, Algebra 1, GeometryRight AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5Vertical AnglesVertical angles are the pair of opposite angles formed when two lines intersect. They share a vertex but no sides, and they are always equal in measure. Two crossing lines make two pairs of vertical angles: if one angle is 120°, the angle across from it is also 120°.Geometry · Grade 7, Grade 8, GeometrySideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometryCounterexampleA counterexample is a single example that shows a general statement is false. It fits all the conditions of the statement but does not have the claimed result. To disprove ‘every prime number is odd,’ one counterexample is enough: 2 is prime and even. Examples that agree with a claim never prove it, but one counterexample disproves it.Geometry · Grade 8, Algebra 1, Geometry