Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 (For help, go to Lesson 2-1 and 2-2.) 1. If it snows tomorrow, then we will go skiing. 2. If two lines are parallel, then they do not intersect. 3. If x = –1, then x2 = 1. Write two conditional statements that make up each biconditional. 4. A point is on the bisector of an angle if and only if it is equidistant from the sides of the angle. 5. A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment. 6. You will pass a geometry course if and only if you are successful with your homework. Write the converse of each statement. 5-4

Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 Solutions 1. Switch the hypothesis and conclusion: If we go skiing tomorrow, then it snows. 2. Switch the hypothesis and conclusion: If two lines do not intersect, then they are parallel. 3. If x2 = 1, then x = –1. 4. Rewrite the statement as an if-then statement; then rewrite it by writing its converse: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. If a point is equidistant from the sides of an angle, then it is on the bisector of the angle. 5. Rewrite the statement as an if-then statement; then rewrite it by writing its converse: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. 6. Rewrite the statement as an if-then statement; then rewrite it by writing its converse: If you pass a geometry course, then you are successful with your geometry homework. If you are successful with your geometry homework, then you will pass the geometry course. 5-4

5-4: Inverses, Contrapositives, and Indirect Reasoning Objective: Learn to write the negation of a statement and the inverse and contrapositive. To use indirect reasoning. Vocabulary: Negation: Has opposite truth value of a given statement. Inverse: Negates both hypothesis and conclusion of a conditional Contrapositive: The inverse of the converse of a conditional. Equivalent Statements: Statements with the same truth value.

Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 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.” 5-4

Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 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. Converse: If ABC is isosceles, then it is equilateral. Contrapositive: If ABC is not isosceles, then it is not equilateral. 5-4

Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 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.” 5-4

Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 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 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 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. 5-4

Inverses, Contrapositives, and Indirect Reasoning GEOMETRY LESSON 5-4 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. 5-4