This document provides an overview of indirect proofs in mathematics. It defines indirect reasoning, indirect proof, and proof by contradiction as forms of proof that assume the opposite of what is being proved in order to derive a contradiction. The document gives examples of stating assumptions for indirect proofs, writing indirect proofs algebraically and geometrically, and identifying the key steps of writing an indirect proof by first assuming the opposite, then deriving a contradiction. It provides fully worked examples of indirect proofs for algebraic, number theory, and geometric statements. Finally, it assigns practice problems for readers to work.

1. Section 5-4
Indirect Proof

2. Essential Questions
How do you write indirect algebraic proofs?
How do you write indirect geometric proofs?

3. Vocabulary
1. Indirect Reasoning:
2. Indirect Proof:
3. Proof by Contradiction:

4. Vocabulary
1. Indirect Reasoning: Assuming that the conclusion
of a conditional is false, then showing this
assumption leads to a contradiction
2. Indirect Proof:
3. Proof by Contradiction:

5. Vocabulary
1. Indirect Reasoning: Assuming that the conclusion
of a conditional is false, then showing this
assumption leads to a contradiction
2. Indirect Proof: Prove a statement to be true by
assuming that which you are trying to prove to
be false, then show that it leads to a
contradiction
3. Proof by Contradiction:

6. Vocabulary
1. Indirect Reasoning: Assuming that the conclusion
of a conditional is false, then showing this
assumption leads to a contradiction
2. Indirect Proof: Prove a statement to be true by
assuming that which you are trying to prove to
be false, then show that it leads to a
contradiction
3. Proof by Contradiction: Another name for indirect
proof

7. Steps to Write an Indirect
Proof

8. Steps to Write an Indirect
Proof
1. Identify what you are trying to prove, then
assume the conclusion to be false (opposite).

9. Steps to Write an Indirect
Proof
1. Identify what you are trying to prove, then
assume the conclusion to be false (opposite).
2. Follow a logical argument based on your false
assumption to find a contradiction in your new
assumption.

10. Steps to Write an Indirect
Proof
1. Identify what you are trying to prove, then
assume the conclusion to be false (opposite).
2. Follow a logical argument based on your false
assumption to find a contradiction in your new
assumption.
3. Identify the contradiction, thus proving that the
original conclusion must be true.

11. Example 1
State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
b. 3x = 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is
congruent to LB.

12. Example 1
State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
EF is a perpendicular bisector.
b. 3x = 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is
congruent to LB.

13. Example 1
State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
EF is a perpendicular bisector.
b. 3x = 4y + 1 3x ≠ 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is
congruent to LB.

14. Example 1
State the assumption necessary to start an indirect
proof for each statement.
a. EF is not a perpendicular bisector.
EF is a perpendicular bisector.
b. 3x = 4y + 1 3x ≠ 4y + 1
c. If B is the midpoint of LH and LH = 26, then BH is
congruent to LB.
BH is not congruent to LB.

15. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.

16. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.

17. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7

18. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7

19. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7

20. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true

21. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7

22. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7

23. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
9 < 7

24. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
9 < 7
Also not true

25. Example 2
Write an indirect proof to show that if −2x + 11 < 7,
then x > 2.
Assume that x ≤ 2.
Then −2(2) + 11 < 7
−4 + 11 < 7
7 < 7
Not true
Also, −2(1) + 11 < 7
−2 + 11 < 7
9 < 7
Also not true
Both cases provide a contradiction, so therefore x > 2
must be true.

26. Example 3
Write an indirect proof to show that if x is a prime
number not equal to 3, then x/3 is not an integer.

27. Example 3
Write an indirect proof to show that if x is a prime
number not equal to 3, then x/3 is not an integer.
Assume x/3 is an integer.

28. Example 3
Write an indirect proof to show that if x is a prime
number not equal to 3, then x/3 is not an integer.
Assume x/3 is an integer.
Then,
x
3
= n.

29. Example 3
Write an indirect proof to show that if x is a prime
number not equal to 3, then x/3 is not an integer.
Assume x/3 is an integer.
Then,
x
3
= n.
This means x = 3n.

30. Example 3
Write an indirect proof to show that if x is a prime
number not equal to 3, then x/3 is not an integer.
Assume x/3 is an integer.
Then,
x
3
= n.
This means x = 3n.
This cannot be true, because if x is prime, then 3
cannot be a factor of x as shown in the equation
x = 3n. By contradiction, x/3 is not an integer.

31. Example 4
Prove by contradiction.
K
J L
8 5
7
Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L

32. Example 4
Prove by contradiction.
K
J L
8 5
7
Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.

33. Example 4
Prove by contradiction.
K
J L
8 5
7
Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Then, by angle-side relationships, JL ≥ JK.

34. Example 4
Prove by contradiction.
K
J L
8 5
7
Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Then, by angle-side relationships, JL ≥ JK.
This mean that 7 ≥ 8, which is not true.

35. Example 4
Prove by contradiction.
K
J L
8 5
7
Given: ∆JKL with side lengths 5, 7, and 8.
Prove: m∠K < m∠L
Assume m∠K ≥ m∠L.
Then, by angle-side relationships, JL ≥ JK.
This mean that 7 ≥ 8, which is not true.
So the assumption m∠K ≥ m∠L is false, so by
contradiction, m∠K < m∠L

36. Problem Set

37. Problem Set
p. 354 #1-29 odd, 43, 51, 53
“It's not that I'm so smart, it's just that I stay with
problems longer.” – Albert Einstein