Indirect Proof

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

Definition

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

Example

Solution

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 AMMB.

AMMB

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.