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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? 1.Triangle PQR is equilateral 2.Triangle PQR is a right triangle 3.Triangle PQR is isosceles

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Identify the pair of statements that form a Contradiction. 1.Triangle PQR is equilateral 2.Triangle PQR is a right triangle 3.Triangle PQR is isosceles 1 & 2

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Identify the pair of statements that form a Contradiction. 1.ABCD is a parallelogram. 2.ABCD is a trapezoid. 3.ABCD has two acute angles.

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Identify the pair of statements that form a Contradiction. 1.ABCD is a parallelogram. 2.ABCD is a trapezoid. 3.ABCD has two acute angles. 1 & 2

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Identify the pair of statements that form a Contradiction. 1.Line l and m are skew. 2.Line l and m do not intersect 3.Line l is parallel to line m.

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Identify the pair of statements that form a Contradiction. 1.Line l and m are skew. 2.Line l and m do not intersect 3.Line l is parallel to line m. 1 & 3

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Identify the pair of statements that form a Contradiction. 1.Segment FG is parallel to segment KL. 2.Segment FG is perpendicular to segment KL. 3.Segment FG is parallel to segment KL.

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Identify the pair of statements that form a Contradiction. 1.Segment FG is parallel to segment KL. 2.Segment FG is perpendicular to segment KL. 3.Segment FG is parallel 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.

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Examples: Step One 1.

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Examples: Step One 1. Segment YX is congruent to segment AB.

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Examples: Step One 1.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 1.Triangle PEN is isosceles. Assume Triangle PEN is scalene.

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Examples: Step One 1. m 90

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Examples: Step One 1.m 90 Assume m<2 90.

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Examples: Step One 1. At least one angle is obtuse

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Examples: Step One 1.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 1.What is the contradiction with step one? Statement: m 90 Step One: Assume m<2 90. Step Two: ? 100

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Examples: Step Two 1.What is the contradiction with step one? Statement: m 90 Step One: Assume m<2 90. Step Two: The m<2 = 110 which is bigger than

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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 cant 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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