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Inverses, Contrapositives, and Indirect Reasoning

1. Complete the sentence: To find the centroid of a triangle, you need to draw at least ? median(s). 2. FGH has vertices F (–1, 2), G (9, 2), and H (9, 0). Find the center of the circle that circumscribes FGH . Use the diagram for Exercises 3–5.

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Inverses, Contrapositives, and Indirect Reasoning

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  1. 1. Complete the sentence: To find the centroid of a triangle, you need to draw at least ? median(s). 2.FGH has vertices F(–1, 2), G(9, 2), and H(9, 0). Find the center of the circle that circumscribes FGH. Use the diagram for Exercises 3–5. 3. Identify all medians and altitudes drawn in PSV. 4. If SY = 15, find SM and MY. 5. If MX = 14, find PM and PX. PX and SY are medians; VZ is an altitude. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Lesson 5-3 Quiz – Concurrent Lines, Medians, and Altitudes two (4, 1) SM = 10 and MY = 5 PM = 28 and PX = 42 5-3

  2. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Remember! 5-4

  3. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Conditional and Related Statements A conditional statement is a statement that can be written in the form “if p, then q.” The converse is the statement formed by exchanging the hypothesis and the conclusion. The negation of statement p is “not p.” The inverse is the statement formed by negating the hypothesis and conclusion. The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. 5-4

  4. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 The negation of a true statement is false, and the negation of a false statement is true. Remember! A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis is true and the conclusion is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false. 5-4

  5. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Related conditional statements that have the same truth value are called logically equivalent statements. A conditional and its contrapositive are logically equivalent, and so are the converse and inverse. 5-4

  6. Helpful Hint The logical equivalence of a conditional and its contrapositive is known as the Law of Contrapositive. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 5-4

  7. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction. 5-4

  8. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 5-4

  9. Helpful Hint When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 5-4

  10. Write the negation of “ABCD is not a convex polygon.” Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Additional Examples Writing the Negation of a Statement The negation of a statement has the opposite truth value. The negation of isnot 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 5-4

  11. Write the inverse and contrapositive of the conditional statement “If ABC is equilateral, then it is isosceles.” 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. Switch and negate both. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Additional Examples Quick Check Writing the Inverse and Contrapositive 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. Contrapositive:If ABC is not isosceles, then it is not equilateral. 5-4

  12. 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 The First Step of an Indirect Proof 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 5-4

  13. 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 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. 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 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. 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. 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. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Additional Examples Quick Check Identifying Contradictions 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. 5-4

  14. Write an indirect proof. Prove:ABC cannot contain 2 obtuse angles. 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. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Additional Examples Indirect Proof Assume that ABC contains two obtuse angles. Let A and B be obtuse. If A and B are obtuse, mA > 90 and mB > 90, so mA + mB > 180. The assumption in Step 1 must be false. ABC cannot contain 2 obtuse angles. Quick Check 5-4

  15. 1. Write the negation of the statement “ D is a straight angle.” 2. Identify two statements that contradict each other. I. x and y are perfect squares. II. x and y are odd. III. x and y are prime. For Exercises 3–6, use the following statement: If is parallel to m, then 1 and 2 are supplementary. 3. Write the converse. 4. Write the inverse. 5. Write the contrapositive. 6. Write the first step of an indirect proof. D is not a straight angle. If 1 and 2 are supplementary, then is parallel to m. If is not parallel to m, then 1 and 2 are not supplementary. If 1 and 2 are not supplementary, then is not parallel to m. Assume that 1 and 2 are not supplementary. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Lesson Quiz I and III 5-4

  16. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Check Skills You’ll Need 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. (For help, go to Lessons 2-1 and 2-2.) Write the converse of each statement. Check Skills You’ll Need 5-4

  17. Inverses, Contrapositives, and Indirect Reasoning Lesson 5-4 Check Skills You’ll Need 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

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