Method

Two-Column Proof

A two-column proof, or formal proof, is a compact way of showing the reasoning behind a mathematical proof. It consists of two columns, one for the statements and one for the reasons. The reasons can be postulates, theorems, or other mathematical reasoning the reader is assumed to be able to follow without difficulty. For example, consider proving the following statement.

If $M$ is the midpoint of $\overline{A B},$ then $A B = 2 A M .$

Three main steps can be followed when writing a two-column proof.

Write the Given Information

In the first row, write the given statement in the left-hand side column. This statement is given, not derived, so write given in the right-hand column.

0. Statements	0. Reasons
1. $M$ is the midpoint of $\overline{A B}$	1. Given

If possible, draw a diagram that helps to derive the information that will be written in the table. This diagram will not be included in the table, though.

Develop Logical Conclusions

Starting from what is given, develop logical statements that help to prove the desired statement. Since point

M

is the midpoint of

\overline{A B},

it divides the segment into two parts of equal length.

M B = A M

Write this equation in the next row in the statements column. Since this equation came from the definition of a midpoint, write that in the reasons column.

0. Statements	0. Reasons
1. $M$ is the midpoint of $\overline{A B}$	1. Given
2. $M B = A M$	2. Definition of Midpoint

Repeat the Step

2

as Many Times as Necessary

The statements written so far are not enough to reach to the desired conclusion, so continue deriving information and combining it until it points to the desired statement. Remember that postulates, theorems, or other mathematical reasoning can be used.

0. Statements	0. Reasons
1. $M$ is the midpoint of $\overline{A B}$	1. Given
2. $M B = A M$	2. Definition of Midpoint

The Segment Addition Postulate says that the length of a segment equals the sum of the lengths of its parts. Then, the following equation can be derived.

A B = A M + M B

As before, write the equation in the left-hand side and the reason in the right-hand side.

0. Statements	0. Reasons
1. $M$ is the midpoint of $\overline{A B}$	1. Given
2. $M B = A M$	2. Definition of Midpoint
3. $A B = A M + M B$	3. Segment Addition Postulate

Next, use the Substitution Property of Equality to substitute the equation written in the second row into the equation written in the third row.

0. Statements	0. Reasons
1. $M$ is the midpoint of $\overline{A B}$	1. Given
2. $M B = A M$	2. Definition of Midpoint
3. $A B = A M + M B$	3. Segment Addition Postulate
4. $A B = A M + A M$	4. Substitution Property of Equality

The right-hand side of the last equation can be simplified by adding the two terms.

0. Statements	0. Reasons
1. $M$ is the midpoint of $\overline{A B}$	1. Given
2. $M B = A M$	2. Definition of Midpoint
3. $A B = A M + M B$	3. Segment Addition Postulate
4. $A B = A M + A M$	4. Substitution Property of Equality
5. $A B = 2 A M$	5. Simplify

Notice that the last statement is the desired one. Therefore, the proof is done!

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