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# Two-Column Proof

A two-column proof is a compact way of showing the reasoning behind a mathematical proof. It consists of two columns with statements on the left-hand side and the reasoning or justification on the right. The reasons can be postulates, theorems, or other mathematical reasoning the reader is assumed to be able to follow without difficulty.
 Statement Reason

To show how two-column proofs are used, prove the following statement. In the first row, write the given information.

 Statement Reason is the midpoint of Given

The length of a segment is the sum the lengths of the segments it contains. This is called the Segment Addition Postulate. Since is a point on the segment, the length of can be expressed as This is written in the next row.

 Statement Reason is the midpoint of Given Segment Addition Postulate

Point is the midpoint of which means that it divides the segment into two parts of equal length:

 Statement Reason is the midpoint of Given Segment Addition Postulate Definition of a midpoint

By substituting into the expression an equivalent expression of is obtained. This is called the Property of Equality.

 Statement Reason is the midpoint of Given Segment Addition Postulate Definition of a midpoint Property of Equality

Lastly, the new expression can be simplified.

 Statement Reason is the midpoint of Given Segment Addition Postulate Definition of midpoint Property of Equality Simplify

It has now been shown that the length of is and since this was the aim, the proof is now concluded.

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