Method

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
$3$	$3$
$3$	$3$
$3$	$3$
$3$	$3$
$3$	$3$

To show how two-column proofs are used, prove the following statement. $and M is If \overline{A B} is a segment between points A and B, the midpoint of that segment, then A B = 2 A M .$ In the first row, write the given information.

Statement	Reason
$M$ is the midpoint of $\overline{A B}$	Given
$3$	$3$
$3$	$3$
$3$	$3$
$3$	$3$

The length of a segment is the sum the lengths of the segments it contains. This is called the Segment Addition Postulate. Since $M$ is a point on the segment, the length of $\overline{A B}$ can be expressed as $A B = A M + M B .$ This is written in the next row.

Statement	Reason
$M$ is the midpoint of $\overline{A B}$	Given
$A B = A M + M B$	Segment Addition Postulate
$3$	$3$
$3$	$3$
$3$	$3$

Point $M$ is the midpoint of $\overline{A B},$ which means that it divides the segment into two parts of equal length: $M B = A M .$

Statement	Reason
$M$ is the midpoint of $\overline{A B}$	Given
$A B = A M + M B$	Segment Addition Postulate
$M B = A M$	Definition of a midpoint
$3$	$3$
$3$	$3$

By substituting $M B = A M$ into the expression $A B = A M + M B,$ an equivalent expression of $A B$ is obtained. This is called the Property of Equality.

Statement	Reason
$M$ is the midpoint of $\overline{A B}$	Given
$A B = A M + M B$	Segment Addition Postulate
$M B = A M$	Definition of a midpoint
$A B = A M + A M$	Property of Equality
$3$	$3$

Lastly, the new expression can be simplified.

Statement	Reason
$M$ is the midpoint of $\overline{A B}$	Given
$A B = A M + M B$	Segment Addition Postulate
$M B = A M$	Definition of midpoint
$A B = A M + A M$	Property of Equality
$A B = 2 A M$	Simplify

It has now been shown that the length of $\overline{A B}$ is $2 A M,$ and since this was the aim, the proof is now concluded.