Example 1 State the Assumption for Starting an Indirect Proof
State the assumption you would make to start an indirect proof of each statement.
a. mABC = mDBG
b. AB FG
AB is not perpendicular to FG
c. ABC is a right triangle
ABC is not a right triangle
Example 2 Write an Indirect Algebraic Proof
Write an indirect proof to show that if 3x + 6 < 15 then x < 3.
Given: 3x + 6 < 15
Step 1 Indirect Proof:
The negation of x < 3 is x 3. That is, assume that x > 3 or x = 3 is true.
Step 2 Make a table with several possibilities for x given that x > 3 or x = 3.
x 3x + 6
When x > 3, 3x 6 15 and when x = 3, 3x + 6 = 15.
Step 3 In both cases, the assumption leads to the contradiction of a known fact.
Therefore, the assumption that x 3 must be false, which means that x < 3 must
Real-World Example 3 Indirect Algebraic Proof
SALES Frankie bought two shirts for just over $54, before tax. A few weeks later,
her friend, Tanya asked her how much each shirt cost. Frankie could not
remember the individual prices. Use inductive reasoning to show that at least one
of the shirts cost more than $27.
Given: The two shirts cost more than $54.
Prove: At least one of the shirts cost more than $27.
That is, if 2x > 54 then x > 27.
Step 1 Assume that neither shirt cost more than $27. That is, x 27 and y 27.
Step 2 If x 27 and y 27, then x + y 54. That is a contradiction because we know
that the two shirts cost more than $54.
Step 3 The assumption leads to the contradiction of a known fact. Therefore, the
assumption that x 27 and y 27 must be false. Thus, at least one of the shirts
had to have cost more than $27.
Example 4 Indirect Proofs in Number Theory
Write an indirect proof to show that if x an odd integer, then x 2 is an odd integer.
Step 1 Given: x is an odd integer.
Prove: x 2 is an odd integer.
Assume that x 2 is an even integer.
Step 2 x is an odd integer, this means that x = 2k +1 for some integer k.
x2 = (2k + 1)2 Substitution.
= (2k + 1) (2k + 1) Multiply.
= 4k2 + 4k + 1 Simplify.
= 2(2k2 + 2k) + 1 Distributive Property
Step 3 2(k 2 2k) 1 is an odd number. This contradicts the assumption, so it must
be false. So, x 2 is an odd integer.
Example 5 Geometry Proof
Given: ABC is not a right angle.
Prove: mABC 90
Step 1 Assume mABC = 90.
Step 2 If mABC = 90, then by definition of a right angle, ABC is a right angle.
However, this contradicts the given statement.
Step 3 Since the assumption leads to a contradiction, the assumption must be false.
Therefore, mABC 90.