Using Indirect Reasoning

1
Using Indirect Reasoning

• 3 steps to writing an Indirect Proof

2
What conclusion follows from the pair of
statements?

• Triangle PQR is equilateral
• Triangle PQR is a right triangle
• Triangle PQR is isosceles

3
Identify the pair of statements that form a
Contradiction.

• Triangle PQR is equilateral
• Triangle PQR is a right triangle
• Triangle PQR is isosceles
• 1 2

4
Identify the pair of statements that form a
Contradiction.

• ABCD is a parallelogram.
• ABCD is a trapezoid.
• ABCD has two acute angles.

5
Identify the pair of statements that form a
Contradiction.

• ABCD is a parallelogram.
• ABCD is a trapezoid.
• ABCD has two acute angles.
• 1 2

6
Identify the pair of statements that form a
Contradiction.

• Line l and m are skew.
• Line l and m do not intersect
• Line l is parallel to line m.

7
Identify the pair of statements that form a
Contradiction.

• Line l and m are skew.
• Line l and m do not intersect
• Line l is parallel to line m.
• 1 3

8
Identify the pair of statements that form a
Contradiction.

• Segment FG is parallel to segment KL.
• Segment FG is perpendicular to segment KL.
• Segment FG is parallel to segment KL.

9
Identify the pair of statements that form a
Contradiction.

• Segment FG is parallel to segment KL.
• Segment FG is perpendicular to segment KL.
• Segment FG is parallel to segment KL.
• 1 2

10
Step One

• Assume that the opposite of what you want to
  prove is true.

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Step One

• Assume that the opposite of what you want to
  prove is true.

Ex) Statement It is raining outside
12
Step One Indirect Proof

• Assume that the opposite of what you want to
  prove is true.

Ex) Statement It is raining outside Assume It
is NOT raining outside.
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Examples Step One

• 1. ltJ is not a right angle.

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Examples Step One

• ltJ is not a right angle.
• Assume ltJ is a right angle

15
Examples Step One

• 1. Segment YX is congruent to segment AB.

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Examples Step One

• Segment YX is congruent to segment AB.
• Assume Segment YX is not congruent to segment AB.

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Examples Step One

• 1. Triangle PEN is isosceles.

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Examples Step One

• Triangle PEN is isosceles.
• Assume Triangle PEN is scalene.

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Examples Step One

• 1. mlt2 gt 90

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Examples Step One

• mlt2 gt 90
• Assume mlt2 90.

21
Examples Step One

• 1. At least one angle is obtuse

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Examples Step One

• At least one angle is obtuse
• Assume that no angles are obtuse.

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Step Two Indirect Proof

• Use logical reasoning to reach a contradiction of
  an earlier statement, such as the given
  information or a theorem. Then state that the
  assumption you made was false.

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Step Two Indirect Proof

• What is the contradiction of step one?

Ex) Statement It is raining outside Step One It
is not raining outside Step Two The clouds are
out and water is coming out of them.
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Examples Step Two

• What is the contradiction with step one?
• Statement mlt2 gt 90
• Step One Assume mlt2 90.
• Step Two ?

100
26
Examples Step Two

• What is the contradiction with step one?
• Statement mlt2 gt 90
• Step One Assume mlt2 90.
• Step Two The mlt2 110 which is bigger than 90.

100
27
Examples Step Two

• What is the contradiction to step one?
• 2. Triangle PEN is isosceles.
• Step One Assume Triangle PEN is scalene.

P
E
N
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Examples Step Two

• What is the contradiction to step one?
• 2. Triangle PEN is isosceles.
• Step One Assume Triangle PEN is scalene.
• Step Two NP and EN are congruent so PEN cant be
  scalene.

P
E
N
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Step 3 Indirect Proof

• State that what you want to prove must be true.

30
What conclusion follows from the pair of
statements?

• There are three types of drawbridges bascule,
  lift, and swing. This drawbridge does not swing
  or lift.

31
What conclusion follows from the pair of
statements?

• There are three types of drawbridges bascule,
  lift, and swing. This drawbridge does not swing
  or lift.
• Conclusion The bridge is a bascule.

32
What conclusion follows from the pair of
statements?

• If this were the day of the party, our friends
  would be home. No one is home.

33
What conclusion follows from the pair of
statements?

• If this were the day of the party, our friends
  would be home. No one is home.
• Conclusion The party is not today.

34
What conclusion follows from the pair of
statements?

• Every air traffic controller in the world speaks
  English on the job. Sumiko does not speak English.

35
What conclusion follows from the pair of
statements?

• Every air traffic controller in the world speaks
  English on the job. Sumiko does not speak
  English.
• Conclusion Sumiko is not an air traffic
  controller.

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3 Steps to an Indirect Proof

• 1. Assume that the opposite of what you want to
  prove is true.
• 2. Use logical reasoning to reach a contradiction
  of an earlier statement, then state that the
  assumption you made was false.
• 3. State that what you want to prove must be true.