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**Using Indirect Reasoning**

3 steps to writing an Indirect Proof

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**What conclusion follows from the pair of statements?**

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

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**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

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**Identify the pair of statements that form a Contradiction.**

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

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**Identify the pair of statements that form a Contradiction.**

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

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**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.

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**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

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**Identify the pair of statements that form a Contradiction.**

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

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**Identify the pair of statements that form a Contradiction.**

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

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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

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**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. <J is not a right angle.

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**Examples: Step One <J is not a right angle.**

Assume <J is a right angle

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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. m<2 > 90

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Examples: Step One m<2 > 90 Assume m<

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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: m<2 > 90 Step One: Assume m< Step Two: ? 100

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**Examples: Step Two What is the contradiction with step one?**

Statement: m<2 > 90 Step One: Assume m< Step Two: The m<2 = 110 which is bigger than 90. 100

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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. 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 can’t be scalene. P E N

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Step 3: Indirect Proof State that what you want to prove must be true.

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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