Two-Column Proofs

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:** $3x + 5 = 20$ **Prove:** $x = 5$ | # | Statement | Reason | |---|---|---| | 1 | $3x + 5 = 20$ | Given | | 2 | $3x = 15$ | Subtraction Property of Equality | | 3 | $x = 5$ | Division 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: | Reason | What it allows | |---|---| | Addition and Subtraction Property of Equality | If $a = b$, then $a + c = b + c$ and $a - c = b - c$ | | Multiplication and Division Property of Equality | If $a = b$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$ for $c \neq 0$ | | Substitution | If $a = b$, you may replace $a$ with $b$ in any statement | | Distributive Property | $a(b + c) = ab + ac$ | | Reflexive Property | $a = a$ | | Symmetric Property | If $a = b$, then $b = a$ | | Transitive Property | If $a = b$ and $b = c$, then $a = c$ | For segments and angles you also use: - **Segment Addition Postulate:** if $B$ lies between $A$ and $C$ on a line, then $AB + BC = AC$ - **Angle Addition Postulate:** if point $D$ lies inside $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$ - **Definitions:** complementary angles add up to $90^\circ$, supplementary angles add up to $180^\circ$ - **Vertical Angles Theorem:** vertical angles have equal measures

Solving an Equation as a Proof

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

Write the given as statement 1: Statement: $3x + 5 = 20$ = Reason: Given

Subtract 5 from both sides: $3x + 5 - 5 = 20 - 5$, so statement: $3x = 15$ = Reason: Subtraction Property of Equality

Divide both sides by 3: $\frac{3x}{3} = \frac{15}{3}$, so statement: $x = 5$ = Reason: Division Property of Equality

Check that the last statement is the Prove statement: Statement 3 is $x = 5$, exactly what we had to prove = The proof is complete

Answer: 1. $3x + 5 = 20$ (Given); 2. $3x = 15$ (Subtraction Property of Equality); 3. $x = 5$ (Division Property of Equality).

Filling In the Missing Reasons

**Given:** $2(x - 4) = 10$. **Prove:** $x = 9$. The statements are 1. $2(x - 4) = 10$, 2. $2x - 8 = 10$, 3. $2x = 18$, 4. $x = 9$. Find the reason for each statement.

Statement 1 is the starting information: $2(x - 4) = 10$ was given to us = Reason 1: Given

Compare statements 1 and 2: $2(x - 4)$ became $2x - 8$: the 2 was multiplied into the brackets = Reason 2: Distributive Property

Compare statements 2 and 3: $2x - 8 + 8 = 10 + 8$: 8 was added to both sides = Reason 3: Addition Property of Equality

Compare statements 3 and 4: $\frac{2x}{2} = \frac{18}{2}$: both sides were divided by 2 = Reason 4: Division Property of Equality

Answer: Given, Distributive Property, Addition Property of Equality, Division Property of Equality. Check: $2(9 - 4) = 2 \cdot 5 = 10$.

A Segment Proof

Points $A$, $B$, $C$, $D$ lie on a line in this order. **Given:** $AB = CD$. **Prove:** $AC = BD$.

Start with the given: Statement: $AB = CD$ = Reason: Given

Write that $BC$ equals itself: Statement: $BC = BC$ = Reason: Reflexive Property

Add $BC$ to both sides of statement 1: Statement: $AB + BC = CD + BC$ = Reason: Addition Property of Equality

Use segment addition on both sides: Statement: $AB + BC = AC$ and $BC + CD = BD$ = Reason: Segment Addition Postulate

Replace both sums in statement 3: Statement: $AC = BD$ = Reason: Substitution

Answer: $AC = BD$. A numerical check: if $AB = CD = 4$ and $BC = 7$, then $AC = 11$ and $BD = 11$.

Supplementary Angles

**Given:** $\angle 1$ and $\angle 2$ are supplementary, $m\angle 1 = 65^\circ$. **Prove:** $m\angle 2 = 115^\circ$.

Write the given: Statement: $\angle 1$ and $\angle 2$ are supplementary, $m\angle 1 = 65^\circ$ = Reason: Given

Use what supplementary means: Statement: $m\angle 1 + m\angle 2 = 180^\circ$ = Reason: Definition of supplementary angles

Put $65^\circ$ in place of $m\angle 1$: Statement: $65^\circ + m\angle 2 = 180^\circ$ = Reason: Substitution

Subtract $65^\circ$ from both sides: Statement: $m\angle 2 = 115^\circ$ = Reason: Subtraction Property of Equality

Answer: $m\angle 2 = 180^\circ - 65^\circ = 115^\circ$, and every step has its reason.

Proving the Vertical Angles Theorem

Two lines cross and form $\angle 1$, $\angle 2$, $\angle 3$ in a row, so $\angle 1$ and $\angle 3$ are vertical angles. **Given:** $\angle 1$ and $\angle 2$ form a linear pair, $\angle 2$ and $\angle 3$ form a linear pair. **Prove:** $m\angle 1 = m\angle 3$.

Write the given: Statement: $\angle 1, \angle 2$ and $\angle 2, \angle 3$ are linear pairs = Reason: Given

Angles of a linear pair lie on a straight line: Statement: $m\angle 1 + m\angle 2 = 180^\circ$ and $m\angle 2 + m\angle 3 = 180^\circ$ = Reason: Linear Pair Postulate (a linear pair is supplementary)

Both sums equal $180^\circ$, so they equal each other: Statement: $m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$ = Reason: Substitution

Subtract $m\angle 2$ from both sides: Statement: $m\angle 1 = m\angle 3$ = Reason: Subtraction Property of Equality

Answer: Vertical angles always have equal measures. From now on you may use this as a reason: Vertical Angles Theorem.

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!

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

Why: 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 $3x + 5 = 20$ to $3x = 15$ uses the Subtraction Property of Equality (5 was subtracted from both sides).

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

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

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

Mistake: Using the Prove statement as a reason or as statement 1

Why: 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.

Mistake: Doing two moves in one row with one reason

Why: Going from $2(x - 4) = 10$ straight to $2x = 18$ uses two different properties, so one reason cannot cover it.

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

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.

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

Building a Shelf Bracket

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

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

Distances on a Straight Road

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

If town $B$ lies between towns $A$ and $C$, then $AB + BC = AC$.

A two-column proof lists statements on the left and their reasons on the right

It starts from the Given and ends with exactly the statement you were asked to Prove

Every reason is the given, a definition, a property, a postulate or an earlier theorem

Properties of equality: whatever you add, subtract, multiply or divide on one side, you do on the other side too

Substitution, the Reflexive, Symmetric and Transitive Properties and the Distributive Property are also valid reasons

For segments and angles use segment and angle addition, the definitions of complementary ($90^\circ$) and supplementary ($180^\circ$) angles and the Vertical Angles Theorem

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

A: 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.

Q: How do I know which reason to write?

A: 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.

Q: What is the difference between Substitution and the Transitive Property?

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

Q: Can a proof have several correct versions?

A: 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.

Exit Presentation

Two-Column Proofs

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Logic & Sets / Mathematical Proofs

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.

Intermediate25 min

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