Write the negation of “ABCD is not a convex polygon.” Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples 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

To write the inverse of a conditional, negate both the hypothesis and Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Quick Check 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. Hypothesis Conclusion Conditional: 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 and conclusion, 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.

Write the first step of an indirect proof. Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Write 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 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.” Quick Check

Identify the two statements that contradict each other. Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Quick Check 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 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. 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 coplanar, and m PQR = 60. on an angle are coplanar, so these II and III 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

Write an indirect proof. Inverses, Contrapositives, and Indirect Reasoning LESSON 5-4 Additional Examples Write an indirect proof. Prove: ABC cannot contain 2 obtuse angles. 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. 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. Quick Check