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.