Two-Column Proofs

Standard form
A two-column proof is one method for writing a proof of a theorem in Geometry. Unlike a paragraph form proof, a two-column proof looks organized and easy to read. Statements that lead to a conclusion are listed in the left column and the reasons that support each statement at the left are listed in the right column.

The standard form of a two-column proof is shown below.

Given: if-part of a theorem
Prove: then-part of a theorem

Proof:

 Statements Reasons 1.   Statement #1, (usually the given part) 1.   . . . 2.   . . . 2.   . . . 3.   . . . 3.   . . . .    . . . .     . . . .    . . . .     . . . .    . . . .     . . . n.   Last statement, (the Prove-part). n.   . . .

Example
To demonstrate, we will prove the following conditional which deals with an algebraic equation. We will use the Algebraic Properties of Equality in the Reasons column to support the Statements.

Conditional: If 5(x+2) = 20, then x = 4.

In the conditional above the hypothesis or the if-part is 5(x+2) = 20, the conclusion or the then-part is x = 4. The solution is presented below and a diagram is not necessary in this situation.

Given: 5(x+2) = 20
Prove: x = 4

Proof:

Statements
Reasons
1.   5(x+2) = 20  1.   Given
2.   5x+10 = 20  2.   Distributive Property of Equality
3.   5x = 10
3.   Subtraction Property of Equality
4.   x = 2  4.   Division Property of Equality

In writing two-column proofs, write the statements in a logical manner—present them as if you are leading the reader in a step-by-step process from the Given to the Prove. Write them with clarity and justify each statement with reasons using the Given Information, Postulates, Theorems (previously proven), Definitions of Terms (such as isosceles triangles, angle bisectors, etc), and Algebraic Properties.