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Corollary If and only if CONVERSE Proof by Contradiction Axiom Implies Purpose: Lets Define Some Terms!!

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Statement:If Rex is a dog then Rex is a mammal. (True) Converse of a Statement Converse:If Rex is a mammal then Rex is a Dog. (False)

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Consider the sentence below: If the angles of the shape add up to 180 o Then the shape is a triangle. To make the converse statement swap around the parts of the statement in the green boxes above To make the converse statement swap around the parts of the statement in the green boxes above If the shape is a triangle. Then the angles of the shape add up to 180 o This is the converse statement (TRUE) This is the converse statement (TRUE) What is the Converse of a Theorem? What is the Converse of a Theorem?

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Not all converse statements are true. Consider the sentence below: Ifa shape is a square Thenthe angles add up to 360 o Now make the converse statement. If a shape is a square. Then the angles add up to 360 o Can you think of a shape with angles of 360 o which is not a square ? Any closed quadrilateral.

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(1) If a triangle has three equal sides then it has three equal angles. (2 ) If a number is even then the number divides by two exactly. (4) If you have thrown a three and a four then your total score is seven with a die. True False (3) If a shape is a square then the shape has parallel sides. Is the Converse True or False?

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Introducing Indirect Proof: Leinster game? Paul and Mike are driving past the Aviva Stadium. The floodlights are on. Paul: Are Leinster playing tonight? Mike: I dont think so. If a game were being played right now we would see or hear a big crowd but the stands are empty and there isnt any noise. Reductio ad Absurdum: Proof by Contradiction

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Sarah left her house at 9:30 AM and arrived at her aunts house 80 miles away at 10:30 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit. Introducing Indirect Proof

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Proof by Contradiction : Algebra Proof by Contradiction : Algebra

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Proof by Contradiction: Inequalities If Tim buys two shirts for just over 60, can you prove that at least one of the shirts cost more than 30??

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Assume neither shirt costs more than 30 + This is a contradiction since we know Tim spent more than 60 Our original assumption must be false At least one of the shirts had to have cost more than 30 QED x 30 y 30 x + y 60

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Geometry : Proof by Contradiction Triangle ABC has no more than one right angle. Can you complete a proof by contradiction for this statement? Assume A and B are right angles We know A + B + C = C = 0 0 which is a contradiction A and B cannot both be right angles A triangle can have at most one right angle By substitution C = 180 0

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Proof by Contradiction: The Square Root of 2 is Irrational To prove that 2 is irrational Assume the contrary: 2 is rational i.e. there exists integers p and q with no common factors such that: (Square both sides) (Multiply both sides by ) (......its a multiple of 2) (......even 2 = even)

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. This contradicts the original assumption. 2 is irrational Q.E.D. (Divide both sides by 2)

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