Presentation on theme: "Write the negation of “ABCD is not a convex polygon.”"— Presentation transcript:
1 Write the negation of “ABCD is not a convex polygon.” Inverses, Contrapositives, and Indirect ReasoningLESSON 5-4Additional ExamplesWrite the negation of “ABCD is not a convex polygon.”The negation of a statement has the opposite truth value. Thenegation of is not in the original statement removes the word not.The negation of “ABCD is not a convex polygon” is “ABCD is aconvex polygon.”Quick Check
2 To write the inverse of a conditional, negate both the hypothesis and Inverses, Contrapositives, and Indirect ReasoningLESSON 5-4Additional ExamplesQuick CheckWrite the inverse and contrapositive of the conditional statement “If ABC is equilateral, then it is isosceles.”To write the inverse of a conditional, negate both the hypothesis andthe conclusion.Hypothesis ConclusionConditional: If ABC is equilateral, then it is isosceles.Negate both.Inverse: If ABC is not equilateral, then it is not isosceles.To write the contrapositive of a conditional, switch the hypothesis andconclusion, then negate both.Conditional: If ABC is equilateral, then it is isosceles.Switch and negate both.Contrapositive: If ABC is not isosceles, then it is not equilateral.
3 Write the first step of an indirect proof. Inverses, Contrapositives, and Indirect ReasoningLESSON 5-4Additional ExamplesWrite the first step of an indirect proof.Prove: A triangle cannot contain two right angles.In the first step of an indirect proof, you assume as true the negationof what you want to prove.Because you want to prove that a triangle cannot contain two rightangles, you assume that a triangle can contain two right angles.The first step is “Assume that a triangle contains two right angles.”Quick Check
4 Identify the two statements that contradict each other. Inverses, Contrapositives, and Indirect ReasoningLESSON 5-4Additional ExamplesQuick CheckIdentify the two statements that contradict each other.I. P, Q, and R are coplanar.II. P, Q, and R are collinear.III. m PQR = 60Two statements contradict each otherwhen they cannot both be trueat the same time.Examine each pair of statements to see whether they contradict each other.I and IIP, Q, and R arecoplanar andcollinear.Three points that lieon the same line areboth coplanar andcollinear, so thesetwo statements donot contradict eachother.I and IIIcoplanar, andm PQR = 60.on an angle arecoplanar, so theseII and IIIcollinear, andm PQR = 60.If three distinctpoints are collinear,they form a straightangle, so m PQRcannot equal 60.Statements II and IIIcontradict each
5 Write an indirect proof. Inverses, Contrapositives, and Indirect ReasoningLESSON 5-4Additional ExamplesWrite an indirect proof.Prove: ABC cannot contain 2 obtuse angles.Step 1: Assume as true the opposite of what you want to prove. Thatis, assume that ABC contains two obtuse angles. Let A and Bbe obtuse.Step 2: If A and B are obtuse, m A > 90 and m B > 90,so m A + m B > 180.Because m C > 0, this means that m A + m B + m C > 180.This contradicts the Triangle Angle-Sum Theorem, which statesthat m A + m B + m C = 180.Step 3: The assumption in Step 1 must be false. ABC cannotcontain 2 obtuse angles.Quick Check
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