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GEOMETRY HELP Write the negation of “ABCD is not a convex polygon.” The negation of a statement has the opposite truth value. The negation of is not in.

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Presentation on theme: "GEOMETRY HELP Write the negation of “ABCD is not a convex polygon.” The negation of a statement has the opposite truth value. The negation of is not in."— Presentation transcript:

1 GEOMETRY HELP Write the negation of “ABCD is not a convex polygon.” The negation of a statement has the opposite truth value. The negation of is not in the original statement removes the word not. The negation of “ABCD is not a convex polygon” is “ABCD is a convex polygon.” Quick Check Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples

2 GEOMETRY HELP Write 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 and the conclusion. To write the contrapositive of a conditional, switch the hypothesis and conclusion, then negate both. Hypothesis Conclusion Conditional: If ABC is equilateral, then it is isosceles. Negate both. Inverse:If ABC is not equilateral, then it is not isosceles. Conditional: If ABC is equilateral, then it is isosceles. Contrapositive: If ABC is not isosceles, then it is not equilateral. Quick Check Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Switch and negate both.

3 GEOMETRY HELP In the first step of an indirect proof, you assume as true the negation of what you want to prove. Because you want to prove that a triangle cannot contain two right angles, you assume that a triangle can contain two right angles. The first step is “Assume that a triangle contains two right angles.” Write the first step of an indirect proof. Prove: A triangle cannot contain two right angles. Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Quick Check

4 GEOMETRY HELP Two statements contradict each other when they cannot both be true at the same time. Examine each pair of statements to see whether they contradict each other. Identify the two statements that contradict each other. I. P, Q, and R are coplanar. II. P, Q, and R are collinear. III. m PQR = 60 Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Quick Check I and II P, Q, and R are coplanar and collinear. Three points that lie on the same line are both coplanar and collinear, so these two statements do not contradict each other. I and III P, Q, and R are coplanar, and m PQR = 60. Three points that lie on an angle are coplanar, so these two statements do not contradict each other. II and III P, Q, and R are collinear, and m PQR = 60. If three distinct points are collinear, they form a straight angle, so m PQR cannot equal 60. Statements II and III contradict each other.

5 GEOMETRY HELP Step 1: Assume as true the opposite of what you want to prove. That is, assume that ABC contains two obtuse angles. Let A and B be obtuse. Write an indirect proof. Prove: ABC cannot contain 2 obtuse angles. 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 states that m A + m B + m C = 180. Step 3: The assumption in Step 1 must be false. ABC cannot contain 2 obtuse angles. Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Quick Check


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