# Indirect Proof

How to prove the given statement by writing indirect proof (proof by contradiction): definition, 1 example, and its solution.

## Definition

Indirect proof (proof by contradiction)

is another way to prove a statement.

Instead of proving a statement directly,

you show a contradiction

to prove a statement indirectly.

To write an indirect proof:

Assume that [~Prove] is true.

Then show a contradiction.

The contradiction is made by the wrong assumption:

[~Prove].

So [~Prove] is false.

And [Prove] is true.

Negation Statement

Two-Column Proof

## Example

Make a two-column form like this.

Name the left column Statement.

And name the right column Reason.

Assume that [~Prove] is true.

Prove: M is not the midpoint of AB.

So write [~Prove]:

M is the midpoint of AB.

Starting from this assumption,

find a contradiction.

M is the midpoint of AB.

The midpoint M divides the segment

into two congruent segments.

So AM ≅ MB.

AM ≅ MB

Then, by the definition of congruent segments,

AM = MB.

See the given statement:

AM ≠ MB.

AM = MB

AM ≠ MB

These two statements show a contradiction.

The contradiction is made by the wrong assumption:

M is the midpoint of AB.

The assumption is false.

So its negation,

M is not the midpoint of AB

is true.

You showed that

M is the midpoint of AB

is true.

This is the [Prove].

So close the two-column form

by drawing the bottom line.

This is the indirect proof

of the given example.