Key takeaway: A two-column proof is a logical argument written as a numbered list of statements in the left column, each justified by a reason in the right column, ending with the claim to be proved.
Grade 10 – GeometryG-CO.C.9Geometry and Measurement
Definition of Two-Column Proof
A two-column proof organizes a geometry argument into rows. The left column holds statements, one claim per line. The right column gives the reason that makes each claim certain.
The first rows usually restate the given information. Later rows apply definitions, postulates, theorems or algebra properties, and every row must follow from rows above it. The last row is exactly what you were asked to prove. A reason can never be “it looks true” or “from the diagram” unless the diagram marks it.
Notation
Every two-column proof uses the same layout, a table with numbered rows:
| Statements | Reasons |
|---|---|
| 1. first claim | Given |
| 2. next claim | definition, theorem or property |
Write “Given” for facts supplied by the problem, and “Prove” for the final target.
Key facts
- Each statement needs its own reason, and each reason must be an accepted fact.
- Start from the given information and end with the statement to prove.
- A proof may use algebra properties such as substitution and the addition property.
- Different valid proofs of the same fact can have different numbers of rows.
Two-Column Proof example
Given: \(M\) is the midpoint of \(\overline{AB}\), with \(AM = 3x + 2\) and \(MB = 5x – 10\). Prove that \(AB = 40\).
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1. \(M\) is the midpoint of \(\overline{AB}\). Reason: Given.
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2. \(AM = MB\). Reason: Definition of midpoint.
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3. \(3x + 2 = 5x – 10\). Reason: Substitution. Solving gives \(x = 6\) (Algebra), so \(AM = 20\).
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4. \(AB = AM + MB = 20 + 20 = 40\). Reason: Segment Addition Postulate.
The last row reads \(AB = 40\), which is the statement to be proved.
Common mistake
Watch out
In a two-column proof, a student writes “\(AM = MB\)” in row 2 and gives the reason “it looks equal in the picture.”
Why it is wrong
A diagram can suggest a fact, but it cannot justify one. The row needs a reason from the given list: here, the definition of a midpoint, because the Given says \(M\) is the midpoint. Every row must cite a given, a definition, a postulate, a theorem or an algebra property.
Related terms and lessons
- Geometry · Logic and Two-Column ProofsLessonWorksheetQuiz
- Geometry · Parallel Lines, Transversals and ProofsLessonWorksheetQuiz
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