Two-Column Proof
In short: A two-column proof lists each logical claim in a left column and the reason that justifies it in a right column, row by row, until the claim to be proved is reached.
GeometryGeometryG-CO.C.9
What is Two-Column Proof?
A two-column proof is a tidy format for an argument. Every row holds a statement on the left, such as a measure or a relationship, and a reason on the right: a given fact, a definition, a postulate or a theorem already proven.
Rows run in order from the givens to the Prove line with no jumps. Each statement must follow from earlier rows, so a reader can test the chain one link at a time, much like checking the steps of a recipe.
How to spot it
- A problem hands you Given and Prove lines and asks you to show why the second follows from the first.
- A table with two headings, Statements and Reasons, has numbered rows waiting to be filled: a classic two-column proof layout.
- A figure carries tick marks and arcs that tell which sides or angles are equal.
- Words such as prove, show that or justify each step appear in the directions.
The method
- Copy the Given and Prove information, then mark the figure with every fact you know.
- Write row 1 from the givens, with Given as the reason.
- Add one row at a time; in a two-column proof every reason is a definition, postulate, theorem or property.
- End with the Prove statement, then read down the table to make sure no row skips a step.
Two-Column Proof: solved example
Problem
Given: \(B\) is the midpoint of \(\overline{AC}\), \(AB = 3x + 2\) and \(BC = 5x – 6\). Prove: \(AC = 28\).
Solution
| Statement | Reason |
|---|---|
| 1. \(B\) is the midpoint of \(\overline{AC}\) | Given |
| 2. \(AB = BC\) | Definition of midpoint |
| 3. \(3x + 2 = 5x – 6\) | Substitution |
| 4. \(x = 4\) | Properties of equality |
| 5. \(AB = 14\) and \(BC = 14\) | Substitution |
| 6. \(AC = 14 + 14 = 28\) | Segment addition postulate |
The chain ends exactly at the Prove line: \(AC = 28\).
Try it
Question 1
Which reason justifies going from \(AB = BC\) and \(BC = 9\) to \(AB = 9\)?
Show the answer
The transitive property of equality, also usable as substitution.
Question 2
From \(\angle A \cong \angle B\), what reason supports the next row \(m\angle A = m\angle B\)?
Show the answer
The definition of congruent angles: congruent angles have equal measures.
Two-Column Proof vs. Conditional Statement
A conditional statement is only a claim: if the givens hold, the result holds. The proof is the argument that shows that claim is true. The claim supplies the Given and Prove lines, while the table of rows is the evidence behind it.
Keep going
Lessons
Worksheets and quizzes
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
- Congruent Triangles and Rigid Motions – Worksheet
- Congruent Triangles and Rigid Motions – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry