Two-Column Algebraic Proofs | Geometry | LevelKite

You already know how to justify each step when you solve an equation. A two-column proof puts that same thinking into a table. A proof is an argument that shows a claim must be true, using only facts you can name.

Every proof begins with two labels:

Given: the facts you are told are true.
Prove: the claim you must show.

The table has two columns. The Statement column holds each claim. The Reason column says why that claim is true. Each step gets one row, with the Given first and the Prove last.

Key concept

Every statement needs a named reason

A two-column proof is a chain from the Given to the Prove. Each statement after the Given must follow from the rows above it. Every reason is the word Given or the exact name of a definition, a postulate (a fact accepted without proof), a property of equality or congruence, or a theorem (a statement already proved). A step can never rest on its own conclusion, on an unstated fact or on how a figure looks.

To complete a proof, you fill in the statements or reasons that are missing. The rows around each blank give you clues:

If a reason is missing, compare its statement with the rows above it. Ask what changed, then name the fact that allows that change.
If a statement is missing, read its reason and apply it to the rows above it.

Try this on a proof about an equation. Given: $5x - 3 = 2x + 12$ . Prove: $x = 5$ .

A proof to complete, with three blanks

Statement	Reason
$5x - 3 = 2x + 12$	Given
$3x - 3 = 12$	Blank 1
Blank 2	Addition Property of Equality
$x = 5$	Blank 3

Worked example

Complete an algebraic proof

Given: $5x - 3 = 2x + 12$ . Prove: $x = 5$ . Fill in Blanks 1, 2 and 3 in the table above.

Start at the top. This equation is what you are told, so its reason is Given.
$5x - 3 = 2x + 12$
Blank 1 is a missing reason, so compare this row with the row above. The 2x is gone from the right side, and 5x became 3x. You subtracted 2x from each side, so the reason is Subtraction Property of Equality.
$3x - 3 = 12$
Blank 2 is a missing statement, so use the reason shown beside it. Addition Property of Equality tells you the same number was added to each side. Adding 3 removes the -3 on the left and turns 12 into 15.
$3x = 15$
Blank 3 is a missing reason. The 3 multiplying x is gone, so you divided each side by 3, which is allowed because 3 is not 0. The reason is Division Property of Equality. This row matches the Prove, so the proof is finished.
$x = 5$

Answer

Blank 1 is Subtraction Property of Equality, Blank 2 is $3x = 15$ and Blank 3 is Division Property of Equality, so the proof shows that $x = 5$ .

Now the variable helps describe a length, so the proof needs a geometry fact. $AB$ means the length of the segment from $A$ to $B$ . The Segment Addition Postulate says: if $B$ is between $A$ and $C$ , then $AB + BC = AC$ . Here between means the three points lie on one line and $B$ sits on the part from $A$ to $C$ .

Given: $B$ is between $A$ and $C$ , $AB = 2x + 3$ , $BC = 10$ and $AC = 19$ . Prove: $x = 3$ .

In the first proof, $x$ was only a number. Here it describes a length, so one more job comes after the last row: check the value. A segment length must be positive, and an angle measure must be greater than 0 and at most 180 degrees. Reject any value that breaks these conditions, such as one that makes a length or a measure zero or negative.

A proof to complete, with three blanks

Statement	Reason
$B$ is between $A$ and $C$	Given
$AB = 2x + 3$	Given
$BC = 10$	Given
$AC = 19$	Given
$AB + BC = AC$	Blank 1
Blank 2	Substitution Property of Equality
$2x + 3 = 9$	Blank 3
$2x = 6$	Subtraction Property of Equality
$x = 3$	Division Property of Equality

Worked example

Complete a proof about a length

Given: $B$ is between $A$ and $C$ , $AB = 2x + 3$ , $BC = 10$ and $AC = 19$ . Prove: $x = 3$ . Fill in Blanks 1, 2 and 3 in the table above, then check the value of $x$ .

The first four rows copy the Given, so each one has the reason Given. One of them says that B is between A and C, which the next row needs.
$AB = 2x + 3,\quad BC = 10,\quad AC = 19$
Blank 1 is a missing reason. This row adds two parts to make the whole, and B is between A and C, so the reason is Segment Addition Postulate.
$AB + BC = AC$
Blank 2 is a missing statement. Its reason is Substitution Property of Equality, so replace each length in the row above with its value from the Given rows.
$(2x + 3) + 10 = 19$
Blank 3 is a missing reason. The 10 is gone from the left side, and 19 became 9. You subtracted 10 from each side, so the reason is Subtraction Property of Equality.
$2x + 3 = 9$
This row is already filled in. Its reason, Subtraction Property of Equality, matches subtracting 3 from each side.
$2x = 6$
This row is filled in too. Its reason, Division Property of Equality, matches dividing each side by 2, which is not 0. The row matches the Prove.
$x = 3$
Check the value. With x = 3, AB is 9, and BC = 10 and AC = 19 are given. All three lengths are positive, so you keep x = 3.
$AB = 2(3) + 3 = 9 > 0$
See what a failed check looks like. If AC were 9 instead of 19, the same steps would give x = -2. Then AB would be negative, and a length must be positive, so you would reject x = -2.
$AB = 2(-2) + 3 = -1 < 0$

Answer

Blank 1 is Segment Addition Postulate, Blank 2 is $(2x + 3) + 10 = 19$ and Blank 3 is Subtraction Property of Equality. The proof shows that $x = 3$ , and this value gives positive lengths.

Now you will write a whole proof yourself. The goal is a geometric equation, an equation about lengths or angle measures. Before you write any rows, make a plan: ask what the Prove needs, then look for named facts that connect it to the Given.

Given: Points $A$ , $B$ , $C$ and $D$ lie on one line in that order, and $\overline{AB} \cong \overline{CD}$ . Prove: $AC = BD$ .

The symbol $\cong$ means is congruent to. The Definition of congruent segments says that congruent segments have equal lengths, and that segments with equal lengths are congruent. Matching tick marks show segments that are given as congruent. In the figure, one tick on segment $AB$ and one on segment $CD$ mark this Given. The figure is not to scale, so the facts come from the Given and the marks, not from how the drawing looks.

One tick on each of segments

AB

and

CD

marks the Given

\overline{AB} \cong \overline{CD}

.

Worked example

Write a proof about segment lengths

Given: Points $A$ , $B$ , $C$ and $D$ lie on one line in that order, and $\overline{AB} \cong \overline{CD}$ . Prove: $AC = BD$ .

Make a plan. The Segment Addition Postulate writes AC as AB + BC and writes BD as BC + CD. Both sums contain BC. So start from AB = CD, add BC to each side, then replace each sum.
Write the first Given. You need the order of the points, because the Segment Addition Postulate requires a point between two others. Reason: Given.
$A,\ B,\ C,\ D \text{ lie on one line in that order}$
Write the second Given. Reason: Given.
$\overline{AB} \cong \overline{CD}$
Change the congruence into an equation. You are using the direction from congruent to equal lengths. Reason: Definition of congruent segments.
$AB = CD$
Add BC to each side. The order of a sum does not change its value, so on the right you may write BC first, which matches the postulate in a later row. Reason: Addition Property of Equality.
$AB + BC = BC + CD$
The order of the points puts B between A and C. Reason: Segment Addition Postulate.
$AB + BC = AC$
The order of the points also puts C between B and D. Reason: Segment Addition Postulate.
$BC + CD = BD$
In the equation AB + BC = BC + CD, replace each sum with the single length it equals. This row matches the Prove, so the proof is finished. Reason: Substitution Property of Equality.
$AC = BD$

Answer

Because $\overline{AB} \cong \overline{CD}$ and the points lie in the order $A$ , $B$ , $C$ , $D$ , the proof shows that $AC = BD$ .

The finished proof about segment lengths

Statement	Reason
$A$ , $B$ , $C$ , $D$ lie on one line in that order	Given
$\overline{AB} \cong \overline{CD}$	Given
$AB = CD$	Definition of congruent segments
$AB + BC = BC + CD$	Addition Property of Equality
$AB + BC = AC$	Segment Addition Postulate
$BC + CD = BD$	Segment Addition Postulate
$AC = BD$	Substitution Property of Equality

Now try an angle proof that also solves for a value. The notation $m\angle AEC$ means the measure of angle $AEC$ , in degrees. Vertical angles are two angles formed by two crossing lines that sit across from each other and share only the crossing point. The Vertical Angles Theorem says: if two angles are vertical angles, then they are congruent. Use it in that direction only, because congruent angles do not have to be vertical angles. The Definition of congruent angles says that congruent angles have equal measures, and that angles with equal measures are congruent.

Given: $\angle AEC$ and $\angle BED$ are vertical angles, $m\angle AEC = 5x - 30$ and $m\angle BED = 3x + 10$ . Prove: $x = 20$ .

In the figure, segments $AB$ and $CD$ cross at $E$ . Angle $AEC$ measures $(5x - 30)^\circ$ and angle $BED$ measures $(3x + 10)^\circ$ . No mark says these angles are equal, and the drawing cannot tell you. The theorem gives you that fact.

Angles

AEC

and

BED

are the vertical angles named in the Given.

Worked example

Write a proof that solves for an angle value

Given: $\angle AEC$ and $\angle BED$ are vertical angles, $m\angle AEC = 5x - 30$ and $m\angle BED = 3x + 10$ . Prove: $x = 20$ .

Plan: turn the vertical angles into an equation in x, then solve it. Begin with the three Given facts. Reason: Given.
$\angle AEC,\ \angle BED \text{ are vertical};\ m\angle AEC = 5x - 30;\ m\angle BED = 3x + 10$
These angles are vertical angles, so they are congruent. Reason: Vertical Angles Theorem.
$\angle AEC \cong \angle BED$
Congruent angles have equal measures. Reason: Definition of congruent angles.
$m\angle AEC = m\angle BED$
Replace each measure with its expression from the Given. Reason: Substitution Property of Equality.
$5x - 30 = 3x + 10$
Subtract 3x from each side. Reason: Subtraction Property of Equality.
$2x - 30 = 10$
Add 30 to each side. Reason: Addition Property of Equality.
$2x = 40$
Divide each side by 2, which is not 0. This row matches the Prove. Reason: Division Property of Equality.
$x = 20$
Check the value. Both angles measure 70 degrees, which is greater than 0 and at most 180, so you keep x = 20.
$5(20) - 30 = 70,\quad 3(20) + 10 = 70$

Answer

The proof shows that $x = 20$ , and this value gives both angles a measure of $70^\circ$ , which meets the angle conditions.

The finished proof about vertical angles

Statement	Reason
$\angle AEC$ and $\angle BED$ are vertical angles	Given
$m\angle AEC = 5x - 30$	Given
$m\angle BED = 3x + 10$	Given
$\angle AEC \cong \angle BED$	Vertical Angles Theorem
$m\angle AEC = m\angle BED$	Definition of congruent angles
$5x - 30 = 3x + 10$	Substitution Property of Equality
$2x - 30 = 10$	Subtraction Property of Equality
$2x = 40$	Addition Property of Equality
$x = 20$	Division Property of Equality

A two-column proof is a chain of named reasons from the Given to the Prove. To complete one, use the rows around each blank to find what is missing. To write one, plan first, then pair each statement with the exact name of its reason. When you solve for a value, check that every length is positive and every angle measure is greater than 0 and at most 180 degrees.