Using Indirect Reasoning

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Warm Up 1. Write a conditional from the
sentence An isosceles triangle has two congruent
sides. 2. Write the contrapositive of the
conditional If it is Tuesday, then John has a
piano lesson. 3. Show that the conjecture If
x gt 6, then 2x gt 14 is false by finding a
counterexample.
If a ? is isosc., then it has 2 ? sides.
If John does not have a piano lesson, then it is
not Tuesday.
x 7
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Using Indirect Reasoning

• 4.5
• Objective Write indirect proofs

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So far you have written proofs using direct
reasoning. You began with a true hypothesis and
built a logical argument to show that a
conclusion was true. In an indirect proof, you
begin by assuming that the conclusion is false.
Then you show that this assumption leads to a
contradiction. This type of proof is also called
a proof by contradiction.
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Definitions

• Indirect Reasoning all possibilities are
  considered and all but one are proven false. The
  remaining possibility must be true.

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Example 1 Writing an Indirect Proof
Step 1 Identify the conjecture to be proven.
Given a gt 0
Step 2 Assume the opposite of the conclusion.
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Example 1 Continued
Step 3 Use direct reasoning to lead to a
contradiction.
Given, opposite of conclusion
Zero Prop. of Mult. Prop. of Inequality
1 ? 0
Simplify.
However, 1 gt 0.
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Example 1 Continued
Step 4 Conclude that the original conjecture is
true.
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Check It Out! Example 1
Write an indirect proof that a triangle cannot
have two right angles.
Step 1 Identify the conjecture to be proven.
Given A triangles interior angles add up to
180.
Prove A triangle cannot have two right angles.
Step 2 Assume the opposite of the conclusion.
An angle has two right angles.
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Check It Out! Example 1 Continued
Step 3 Use direct reasoning to lead to a
contradiction.
m?1 m?2 m?3 180
90 90 m?3 180
180 m?3 180
m?3 0
However, by the Protractor Postulate, a triangle
cannot have an angle with a measure of 0.
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Check It Out! Example 1 Continued
Step 4 Conclude that the original conjecture is
true.
The assumption that a triangle can have two right
angles is false.
Therefore a triangle cannot have two right angles.
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Example I

• Use indirect reasoning to prove this statement
  If Jaelene spends more than 50 to buy two items
  at a bicycle shop, then at least one of the items
  costs more than 25.
• Given The cost of two items is more than 50
• Prove At least one of the items is more than
  25.

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Example I cont

• Begin by assuming that the opposite is true. That
  is assume that neither item is more than 25
• This means both items cost less than 25. This
  then means the two items cost less than 50 which
  contradicts the given information that the amount
  spent is more than 50. So the assumption that
  neither costs more than 25 must be incorrect.
• Therefore, at least one item costs more than 25

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Example II

• Given LMN
• Prove LMN has at most one right angle
• Step 1) Assume LMN has more than one right angle.
  That is ltL and ltM are right angles
• Step 2) If ltL and ltM are both right angles then
  ltL ltM 90 according to the triangle sum
  theorem. Substitution gives us 90 90 ltN 180.
  Solving leaves ltN 0. This means there is no
  angle at N. So the assumption that ltL and ltM are
  both right angles must be false.
• Step 3) Therefore, LMN has at most one right
  angle

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Homework

• Pg 209 1-15, 18-20