Lesson 11-3
Lesson
11-3
Indirect Proof
Vocabulary
direct reasoning
direct proof
indirect reasoning
indirect proof
Indirect proof is based on the idea of reasoning
from a supposition until a contradiction is obtained, and then, as
a result, asserting that the supposition must be false.
contradictory statements
A lawyer, summing up a case for the jury, said “The prosecutors
claim that my client is guilty. Let us suppose, for a moment, that he
is guilty. If my client is guilty, then he must have been at the scene of
the crime when the crime was committed. But as you remember, we
brought in witnesses and telephone records that demonstrate that
my client was 50 miles away at the time. So he wasn’t at the scene of
the crime when it was committed. He could not have been both at the
scene of the crime and also 50 miles away. Our assumption has led to
a contradiction; therefore the supposition must be false and my client
is not guilty.”
Mental Math
BIG IDEA
For each figure, list the
number and type of faces.
a. tetrahedron
b. hexagonal prism
c. frustum of a pyramid
with pentagonal base
In this summation, the lawyer has used indirect reasoning.
In direct reasoning, a person begins with given information known to
be true. The Laws of Detachment and Transitivity are used to reason
from that information to a conclusion. The proofs you have written so
far in this book have been direct proofs.
In indirect reasoning, a person examines and tries to rule out all the
possibilities other than the one thought to be true. This is exactly what
you did in solving the logic puzzles of Lesson 11-1. You marked Xs in
boxes to show which possibilities could not be true. When you had
enough Xs, you knew that the only possibility left must be correct.
The client must have cringed when he heard his lawyer say, “Let
us suppose, for a moment, that he is guilty.” But this is an effective
argument using indirect reasoning. If not-p leads to a contradiction,
not-p must be false. This makes p true. This is the fi fth and last law
of logic discussed in this book.
Law of Indirect Reasoning
If valid reasoning from a statement p leads to a false statement,
then p is false (so not-p is true).
Indirect Proof
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Chapter 11
A proof that uses indirect reasoning is called an indirect proof.
It is helpful to think about an indirect proof as having three parts.
Supposition: Begin by supposing the negation of what you want to
prove. (If you want to prove p, suppose not-p.)
Deduction to Contradiction: Use valid reasoning to show that
the supposition leads to a false statement.
Final Conclusion: Use the Law of Indirect Reasoning to conclude
what you want to prove.
QY
QY
Identify these three parts
of the lawyer’s argument.
Notice that the first part, the supposition, is quite different from the
first step in a direct proof. In a direct proof, the first step usually is
one of the given statements that we know to be true. A supposition, on
the other hand, is just something that is supposed without knowing
whether it is true or false, in the hope that it turns out to be false.
Contradictory Statements
In the deduction part of an indirect proof, you need to show that
a statement is false. But how can you tell if it is false? One way to
tell that a statement is false is if you know its negation is true. For
instance, suppose you know y = 5 is true. Then y ≠ 5 must be false.
Suppose you know ABC is isosceles. Then the statement ABC is
scalene must be false.
You also know a statement is false if it contradicts another statement
known to be true. Two statements p and q are contradictory if and
only if they cannot both be true at the same time. For instance, if
Jayla is a freshman, then she cannot be a junior. If you know x is
positive, then x cannot be negative. If you know a triangle is scalene,
then it cannot be isosceles. A statement and its negation are always
contradictory.
Indirect Proofs You Have Seen
One of the first proofs in this book, for the Line Intersection
Theorem on page 34, was indirect. Here it is reproduced with the
three parts identified.
Theorem Two different lines intersect in at most one point.
Proof
Supposition: Suppose that two lines intersect in
two different points P and Q.
Deduction to Contradiction: Then through P and Q there are
two lines. But the Unique Line Assumption of the Point-Line-Plane
Postulate indicates that there is exactly one line through two points.
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P
Q
Indirect Proofs and Coordinate Proofs
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Lesson 11-3
Final Conclusion: This contradiction indicates that the supposition
is false. So, applying the Law of Indirect Proof, two different lines
cannot intersect in two different points.
When you have encountered an equation or system with no solution,
you may have been doing an indirect proof without knowing it.
Example 1
Show that the system
{5x x++20y4y == 243 has no solution.
Solution Supposition: Suppose there is a solution.
Deduction to Contradiction: Then, to find the solution, multiply both sides
of the second equation by 5.
= 24
{5x5x ++ 20y
20y = 15
Subtracting the second equation from the first,
0 = 9.
This contradicts the fact that 0 ≠ 9.
Final Conclusion: So the supposition must be false. Consequently,
applying the Law of Indirect Reasoning, there is no solution.
A Famous Indirect Proof from Euclid’s Elements
Euclid’s Elements cover more than geometry. Included among the
topics are theorems about prime numbers. Recall that the prime
numbers are those positive integers that are divisible only by
themselves and 1. In order, the prime numbers are 2, 3, 5, 7, 11, 13,
17, 19, 23, . . . Recall that a positive integer greater than 1 that is not
prime is called a composite number.
For the following proof, you need to know two things Euclid had
already proved: First, if a number n is divisible by a particular prime
number, then n + 1 cannot also be divisible by that prime. Second,
every integer has a unique factorization into primes.
Infinitude of Primes Theorem
There are infinitely many prime numbers.
Indirect Proof
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Chapter 11
Proof The proof is indirect.
Supposition: Suppose that there are only finitely many primes. Then
there is a largest prime. Call that number P.
Deduction to Contradiction: Let n be the number that is 1 greater than
the product of all the prime numbers: n = 2 · 3 · 5 · 7 · . . . · P + 1.
Now either n is prime or n is composite. If n is prime, then because it
is obviously larger than P, we have a contradiction of the supposition.
If n is composite, then it is factorable into primes. But it cannot have
any of the primes from 2 to P as a factor because it is 1 greater than a
multiple of them. So it would have to have a prime greater than P as a
factor. This also contradicts the supposition.
Final Conclusion: Because in both cases the supposition leads to a
contradiction, the supposition is false. Applying the Law of Indirect
Proof, there are not finitely many prime numbers. There must be
infinitely many primes.
Still Another Indirect Proof
Here is an example using content from this book.
Example 2
Prove: In a scalene triangle, no median is an altitude.
Solution First set up the given, what is to be proved, and draw a picture
as you would do with a direct proof.
Given: ABC is scalene.
___
CM is a median of ABC.
___
C
___
Prove: CM is not perpendicular to AB.
−−
−−
Supposition: Suppose CM ⊥ AB.
A
Deduction to Contradiction:
Conclusions
B
Justifications
1. ∠AMC ∠CMB
−−
2. CM is a median.
−−
−−
3. AM MB
−−
−−
4. CM CM
1. ⊥ lines ⇒ right ∠s
5. AMC BMC
−−
−−
6. AC BC
5. SAS (Steps 1, 3, 4)
7. ABC is isosceles.
7. definition of isosceles triangle
8. ABC is scalene.
8. Given
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M
2. Given
3. definition of median
4. Reflexive Property of Congruence
6. CPCF Theorem
Indirect Proofs and Coordinate Proofs
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Lesson 11-3
−−
−−
Final Conclusion: Since the supposition CM ⊥ AB leads to
contradictory statements (Steps 7 and 8), it must be false. So,
−−
−−
by the Law of Indirect Reasoning, CM is not perpendicular to AB.
The form of Example 2 can serve as a model for other indirect proofs.
GUIDED
Example 3
Prove that no convex quadrilateral has four acute angles.
Solution
D
Given: Quadrilateral ABCD
Prove:
A
?
?
Proof: Supposition: Suppose ABCD has
.
C
Deduction to Contradiction: Then, by the definition of
?
?
?
, m∠A < 90, m∠B < 90,
< 90, and
.
B
Adding the sides of these four inequalities,
m∠A + m∠B + m∠C + m∠D < 90 + 90 + 90 + 90
m∠A + m∠B + m∠C + m∠D < 360.
But, by the Quadrilateral-Sum Theorem,
?
= 360.
Final Conclusion: Because the supposition leads
to a contradiction, it must be false. Therefore,
?
?
by
,
.
Questions
COVERING THE IDEAS
1. What are the three parts of an indirect proof?
2. a. What is a contradiction?
b. What are contradictory statements?
3. State the Law of Indirect Reasoning.
4. Give an example of a contradiction different from any in
this lesson.
5. In an indirect argument to prove that two coplanar lines are
parallel, what supposition might you start with?
In 6 and 7, statements p and q are given.
a. Are p and q contradictory?
b. Explain your answer.
6. p: JKLM is a kite.
q: JKLM is a rectangle.
___
___
7. p: AB ⊥ BC
q: m∠ABC = 88
Indirect Proof
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Chapter 11
8. Pretend you are about to prove the following theorem with an
indirect proof: If P is a point not on line m, then there is not
more than one line through P parallel to line m. What would
you have to show in the deduction part of the proof?
9. Use an indirect argument, as in the Examples, to show that the
lines with equations 2x - y = 5 and 3y = 6x + 5 are parallel
without using the idea of slope.
10. Refer to the proof that there are infinitely many prime numbers.
a. Explain how you can tell that no prime number less than 13 is
a factor of 2 · 3 · 5 · 7 · 11 + 1.
b. 2 · 3 · 5 · 7 · 11 · 13 + 1 = 59 · 509, and both 59 and 509 are
prime. What does this fact have to do with the proof?
11. Use an indirect proof to prove: In a scalene triangle, no altitude
is a median.
12. a. Prove that no convex quadrilateral has four obtuse angles.
b. Can a convex quadrilateral have three obtuse angles? If so,
draw such a quadrilateral. If not, why not?
13. Use an indirect proof.
Given ABC.
Prove ABC cannot have two right angles.
C
A
B
APPLYING THE MATHEMATICS
14. Bianca just turned 16, the legal age for driving in her state.
Bianca’s father said, “A kid of your age should not be driving.”
Bianca replied, “You may be right.” (She now had his attention.)
Bianca continued, “Look what happens if I don’t drive. I will
not be able to run errands for Mom. I will have to be driven to
school, and the school functions you want me to attend, or not
be able to attend them. This will disrupt both of your schedules.
I think this may contradict what you want to happen.” Bianca
stopped here and let her father come to a conclusion.
a. What conclusion could Bianca’s father make?
b. Identify the three parts of an indirect proof in Bianca’s logic.
1393
15. As a decimal, √
2 = 1.41421 … . As a decimal, ____
= 1.41421 … .
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1393
Show, by indirect reasoning, that √
2 ≠ ____. (Hint: Begin by
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assuming the two numbers are equal. Then square both sides of
the equation.)
16. Use an indirect proof to show that the equation
5(3x - 12) = 3(5x - 18) has no solution.
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Indirect Proofs and Coordinate Proofs
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Lesson 11-3
17. Write an indirect proof that “There is no integer that is greater
than all other integers.” (Hint: If you consider a largest integer
N, can you create a larger one?)
18. Prove that if a rectangle is not a square, its diagonals cannot be
perpendicular.
QY ANSWER
REVIEW
19. Consider the statement: If Jimmy shows up on time, I will eat my
hat! Write its converse, inverse, and contrapositive. (Lesson 11-2)
20. Make a conclusion using all three of the following true
statements. (Lesson 11-2)
(1) Every square is a rhombus.
(2) The diagonals of a kite are perpendicular.
(3) If a figure is a rhombus, then it is a kite.
Supposition (Step 1): The
lawyer supposes his client
was guilty. Deduction to
Contradiction (Step 2): The
lawyer shows that his client
could not have been present
at the scene of the crime at
the time of the crime. Final
Conclusion (Step 3): The
lawyer concludes that his
client is not guilty.
21. Trace the perspective view of the room shown at the
right. (Lesson 9-4)
a. Draw a door on the back wall in perspective.
b. On the right wall, draw a window in perspective.
c. Draw a rectangular rug on the floor in
perspective.
22. Given: ABCD is a parallelogram.
Prove: AOF COE (Lesson 7-7)
23. a. Find the slope of the line with equation 9x + 8y = 36.
b. Give an equation of a line that is perpendicular to the
line from Part a. (Lessons 3-8, 1-2)
E
B
C
O
A
F
D
24. Give a good definition and a bad definition of circle. (Lesson 2-4)
25. Are the points (2, 7), (–3, 14), and (101, 102) collinear? How do
you know? (Lesson 1-2)
26. Explain why (a - b)2 = (b - a)2 for all values of a and b.
(Previous Course)
EXPLORATION
Use ___
27. At the right, points A through G are vertices of a cube.
___
either a direct proof or an indirect proof to show that AC and CE
are not perpendicular.
F
A
E
G
28. Write a story similar to the one in Question 14. Then identify the
parts of an indirect proof in the story you wrote.
D
B
C
Indirect Proof
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