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Teacher Guide: Two-Column Proofs

Learn to write two-column proofs: list each statement on the left and the reason that justifies it on the right, for equations, segments and angles.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Mathematical Proofs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Set up a two-column proof with Given, Prove, numbered statements and reasons
  • Justify each step of solving an equation with a property of equality
  • Write simple proofs about segments and angles using addition postulates and angle definitions
  • Fill in missing statements and reasons in a partly completed proof
Prerequisites
  • • Introduction to Mathematical Proofs: what a proof is and why we need one
  • • Direct Proof: starting from the hypothesis and reasoning forward
  • • Solving one-step and two-step linear equations
  • • Basic angle facts: right angles, straight angles, vertical angles
Discussion Starters
  • 1. Why is "I moved the number to the other side" not a good reason in a proof?
  • 2. Which is easier to check: a proof in two columns or the same argument written as a story? Why?
  • 3. Can a proof be correct if the steps come in a different order than your neighbour's proof?
  • 4. Why is it not allowed to use the Prove statement as a reason?
Common Misconceptions

The reason describes the arithmetic ("20 minus 5 is 15") instead of naming a property

Remediation: Give students a fixed list of allowed reasons. Each reason must come from the list, so they learn to connect each move with its property.

Any true statement can go in the proof, even if it does not follow from the rows above

Remediation: Ask for each row: which earlier rows does it use? Draw arrows from those rows. A row with no arrow into it (other than the given) does not belong.

Undoing a division means using the Division Property

Remediation: Have students write the operation done to both sides in the margin, such as "⋅4\cdot 4". The operation written there names the property.

Differentiation Ideas

For Struggling Students:

  • • Start with proofs where all statements are given and only reasons are missing
  • • Provide a reason bank of 8 to 10 reasons to choose from
  • • Use only one-step and two-step equations before moving to angles

For On-Level Students:

  • • Write complete algebraic proofs with brackets and variables on both sides
  • • Prove segment and angle facts using addition postulates
  • • Find and fix the wrong reason in a flawed proof

For Advanced Students:

  • • Prove the Vertical Angles Theorem without help
  • • Write two different correct proofs for the same Given and Prove
  • • Rewrite a two-column proof as a paragraph proof and compare the two
Standards Alignment
  • HSA-REI.A.1 (CCSS.MATH.CONTENT.HSA.REI.A.1)

    Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step

  • HSG-CO.C.9 (CCSS.MATH.CONTENT.HSG.CO.C.9)

    Prove theorems about lines and angles, including that vertical angles are congruent

  • MP3 (CCSS.MATH.PRACTICE.MP3)

    Construct viable arguments and critique the reasoning of others

Lesson Resources
  • visualStep-by-Step Equation Solver

    Students solve equations step by step and name the property behind each move

  • activityProof Puzzle Strips

    Cut statements and reasons into strips; pairs rebuild the proof in the correct order

  • worksheetMissing Reasons

    Partly finished two-column proofs where students fill in the blank reasons and statements

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A two-column proof is a proof written as a table. The left column lists the statements (what we know at each step) and the right column lists the reasons (why each statement is true). The proof starts from the Given (what we are told) and ends with the statement we were asked to Prove.
Given: 3x+5=203x + 5 = 20 Prove: x=5x = 5
#StatementReason
13x+5=203x + 5 = 20Given
23x=153x = 15Subtraction Property of Equality
3x=5x = 5Division Property of Equality
Every reason must be the given information, a definition, a property, a postulate or a theorem that was proven earlier. The reasons you will use most often:
ReasonWhat it allows
Addition and Subtraction Property of EqualityIf a=ba = b, then a+c=b+ca + c = b + c and a−c=b−ca - c = b - c
Multiplication and Division Property of EqualityIf a=ba = b, then ac=bcac = bc and ac=bc\frac{a}{c} = \frac{b}{c} for c≠0c \neq 0
SubstitutionIf a=ba = b, you may replace aa with bb in any statement
Distributive Propertya(b+c)=ab+aca(b + c) = ab + ac
Reflexive Propertya=aa = a
Symmetric PropertyIf a=ba = b, then b=ab = a
Transitive PropertyIf a=ba = b and b=cb = c, then a=ca = c
For segments and angles you also use:
  • Segment Addition Postulate: if BB lies between AA and CC on a line, then AB+BC=ACAB + BC = AC
  • Angle Addition Postulate: if point DD lies inside ∠ABC\angle ABC, then m∠ABD+m∠DBC=m∠ABCm\angle ABD + m\angle DBC = m\angle ABC
  • Definitions: complementary angles add up to 90∘90^\circ, supplementary angles add up to 180∘180^\circ
  • Vertical Angles Theorem: vertical angles have equal measures

Worked Examples

Given: 3x+5=203x + 5 = 20. Prove: x=5x = 5. Write the proof in two columns.
1

Write the given as statement 1

Statement: 3x+5=203x + 5 = 20 → Reason: Given
2

Subtract 5 from both sides

3x+5−5=20−53x + 5 - 5 = 20 - 5, so statement: 3x=153x = 15 → Reason: Subtraction Property of Equality
3

Divide both sides by 3

3x3=153\frac{3x}{3} = \frac{15}{3}, so statement: x=5x = 5 → Reason: Division Property of Equality
4

Check that the last statement is the Prove statement

Statement 3 is x=5x = 5, exactly what we had to prove → The proof is complete

Common Mistakes

Writing what you did instead of a reason, for example "I moved the 5 to the other side"

Why it's wrong: A reason must name a rule that everyone accepts. "Moving" a number is not a rule, and it hides what really happened.

Correct: Name the property: going from 3x+5=203x + 5 = 20 to 3x=153x = 15 uses the Subtraction Property of Equality (5 was subtracted from both sides).

Choosing Division instead of Multiplication for x4=3\frac{x}{4} = 3 to x=12x = 12

Why it's wrong: The step undoes a division, but the move you make is multiplying both sides by 4.

Correct: Ask what you did to both sides. x4⋅4=3⋅4\frac{x}{4} \cdot 4 = 3 \cdot 4 is the Multiplication Property of Equality.

Using the Prove statement as a reason or as statement 1

Why it's wrong: The Prove statement is the goal. If you use it before you reach it, the proof goes in a circle and proves nothing.

Correct: Statement 1 is always given information. The Prove statement appears only once, as the last statement.

Doing two moves in one row with one reason

Why it's wrong: Going from 2(x−4)=102(x - 4) = 10 straight to 2x=182x = 18 uses two different properties, so one reason cannot cover it.

Correct: One move per row: first 2x−8=102x - 8 = 10 (Distributive Property), then 2x=182x = 18 (Addition Property of Equality).

Why It Matters

Two-column proofs make every step of an argument visible:
  • No hidden jumps: each statement needs its own reason, so a gap or a mistake shows up at once
  • Algebra with understanding: when you solve an equation, you learn exactly which property allows each move
  • Geometry: most school proofs about segments, angles and triangles are written in two columns
  • Programming: checking code line by line, with a reason for every line, works the same way
  • Clear communication: a reader can check your proof one row at a time
Once you can justify every step, you can trust your answer, and so can everyone else!

Real World Applications

Checking Code Line by Line

Programmers prove that a calculation is correct by justifying every line, just like the rows of a two-column proof.

Example:

A program solves 4x−7=214x - 7 = 21. A reviewer checks each line: add 7 (Addition Property), then divide by 4 (Division Property).

1Try It Yourself

The program receives 4x−7=214x - 7 = 21 as given and must prove that x=7x = 7.

Which value does the proof reach for xx?

Step 1: Write the mathematical expression

Write the calculation that undoes both operations:

Building a Shelf Bracket

Carpenters use the Angle Addition Postulate when a diagonal brace splits the right angle between a wall and a shelf.

Example:

The wall and the shelf meet at 90∘90^\circ. A brace splits this corner into two angles that add up to 90∘90^\circ.

2Try It Yourself

The brace makes an angle of 38∘38^\circ with the wall. The two angles at the corner together form the 90∘90^\circ angle between wall and shelf.

Prove the size of the angle between the brace and the shelf.

Step 1: Write the mathematical expression

Write the calculation for the second angle:

Distances on a Straight Road

On a straight road, distances between towns follow the Segment Addition Postulate, so a missing distance can be proven.

Example:

If town BB lies between towns AA and CC, then AB+BC=ACAB + BC = AC.

3Try It Yourself

Towns AA, BB, CC lie on a straight road in this order. AB=35AB = 35 km and AC=120AC = 120 km.

Prove how far it is from BB to CC.

Step 1: Write the mathematical expression

Write the calculation for BCBC:

Key Takeaways

  • 1A two-column proof lists statements on the left and their reasons on the right
  • 2It starts from the Given and ends with exactly the statement you were asked to Prove
  • 3Every reason is the given, a definition, a property, a postulate or an earlier theorem
  • 4Properties of equality: whatever you add, subtract, multiply or divide on one side, you do on the other side too
  • 5Substitution, the Reflexive, Symmetric and Transitive Properties and the Distributive Property are also valid reasons
  • 6For segments and angles use segment and angle addition, the definitions of complementary (90∘90^\circ) and supplementary (180∘180^\circ) angles and the Vertical Angles Theorem

Frequently Asked Questions

Is a two-column proof different from solving an equation?

The steps are the same. The difference is that in a proof you write the reason for every step, so you show not only the answer but also why each move is allowed.

How do I know which reason to write?

Compare the new statement with the one before it and ask what changed. Was something added to both sides? Was a bracket multiplied out? Was a value put in place of another one? The change tells you the reason.

What is the difference between Substitution and the Transitive Property?

Transitive works with three equal things: if a=ba = b and b=cb = c, then a=ca = c. Substitution is more general: you replace a quantity with an equal one inside any statement, for example 65∘65^\circ in place of m∠1m\angle 1. Many teachers accept either one when both fit.

Can a proof have several correct versions?

Yes. You can sometimes do the steps in a different order or use a different property. A proof is correct as long as every statement follows from earlier ones and every reason is valid.

Glossary

Two-column proof
A proof written as a table with statements on the left and reasons on the right
Statement
One step of the proof: a fact we know at that point, such as 3x=153x = 15
Reason
The rule that justifies a statement: given, definition, property, postulate or theorem
Given / Prove
Given is the information you start from; Prove is the statement you must reach at the end
Property of equality
A rule saying that an equation stays true when you do the same operation to both sides
Postulate
A statement accepted without proof, such as the Segment Addition Postulate

Formula Card

Addition Property of Equality

a=b⇒a+c=b+ca = b \Rightarrow a + c = b + c

Add the same number to both sides

Subtraction Property of Equality

a=b⇒a−c=b−ca = b \Rightarrow a - c = b - c

Subtract the same number from both sides

Multiplication Property of Equality

a=b⇒ac=bca = b \Rightarrow ac = bc

Multiply both sides by the same number

Division Property of Equality

a=b⇒ac=bc, c≠0a = b \Rightarrow \frac{a}{c} = \frac{b}{c},\ c \neq 0

Divide both sides by the same nonzero number

Distributive Property

a(b + c) = ab + ac

Multiply a number into brackets

Reflexive, Symmetric, Transitive

a=a; a=b⇒b=a; a=b, b=c⇒a=ca = a;\ a = b \Rightarrow b = a;\ a = b,\ b = c \Rightarrow a = c

Basic properties of equality

Segment Addition Postulate

AB + BC = AC

When B lies between A and C on a line

Complementary and Supplementary Angles

m∠1+m∠2=90∘ or 180∘m\angle 1 + m\angle 2 = 90^\circ \text{ or } 180^\circ

Definitions of complementary and supplementary angles

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