Inductive and Deductive Reasoning 2.2, Exams of Reasoning

Download Inductive and Deductive Reasoning 2.2 and more Exams Reasoning in PDF only on Docsity! Section 2.2 Inductive and Deductive Reasoning 75 Inductive and Deductive Reasoning2.2 Writing a Conjecture Work with a partner. Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern. a. 1 2 3 4 5 6 7 b. c. Using a Venn Diagram Work with a partner. Use the Venn diagram to determine whether the statement is true or false. Justify your answer. Assume that no region of the Venn diagram is empty. a. If an item has Property B, then it has Property A. b. If an item has Property A, then it has Property B. c. If an item has Property A, then it has Property C. d. Some items that have Property A do not have Property B. e. If an item has Property C, then it does not have Property B. f. Some items have both Properties A and C. g. Some items have both Properties B and C. Reasoning and Venn Diagrams Work with a partner. Draw a Venn diagram that shows the relationship between different types of quadrilaterals: squares, rectangles, parallelograms, trapezoids, rhombuses, and kites. Then write several conditional statements that are shown in your diagram, such as “If a quadrilateral is a square, then it is a rectangle.” Communicate Your Answer 4. How can you use reasoning to solve problems? 5. Give an example of how you used reasoning to solve a real-life problem. CONSTRUCTING VIABLE ARGUMENTS To be profi cient in math, you need to justify your conclusions and communicate them to others. Essential Question How can you use reasoning to solve problems? A conjecture is an unproven statement based on observations. Property C Property A Property B hs_geo_pe_0202.indd 75 1/19/15 9:01 AM 76 Chapter 2 Reasoning and Proofs 2.2 Lesson What You Will Learn Use inductive reasoning. Use deductive reasoning. Using Inductive Reasoning Describing a Visual Pattern Describe how to sketch the fourth fi gure in the pattern. Then sketch the fourth fi gure. Figure 1 Figure 2 Figure 3 SOLUTION Each circle is divided into twice as many equal regions as the fi gure number. Sketch the fourth fi gure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left. Figure 4 Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Sketch the fi fth fi gure in the pattern in Example 1. Sketch the next fi gure in the pattern. 2. 3. conjecture, p. 76 inductive reasoning, p. 76 counterexample, p. 77 deductive reasoning, p. 78 Core Vocabulary Core Concept Inductive Reasoning A conjecture is an unproven statement that is based on observations. You use inductive reasoning when you fi nd a pattern in specifi c cases and then write a conjecture for the general case. hs_geo_pe_0202.indd 76 1/19/15 9:01 AM Section 2.2 Inductive and Deductive Reasoning 79 Using Inductive and Deductive Reasoning What conclusion can you make about the product of an even integer and any other integer? SOLUTION Step 1 Look for a pattern in several examples. Use inductive reasoning to make a conjecture. (−2)(2) = −4 (−1)(2) = −2 2(2) = 4 3(2) = 6 (−2)(−4) = 8 (−1)(−4) = 4 2(−4) = −8 3(−4) = −12 Conjecture Even integer • Any integer = Even integer Step 2 Let n and m each be any integer. Use deductive reasoning to show that the conjecture is true. 2n is an even integer because any integer multiplied by 2 is even. 2nm represents the product of an even integer 2n and any integer m. 2nm is the product of 2 and an integer nm. So, 2nm is an even integer. The product of an even integer and any integer is an even integer. Comparing Inductive and Deductive Reasoning Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. a. Each time Monica kicks a ball up in the air, it returns to the ground. So, the next time Monica kicks a ball up in the air, it will return to the ground. b. All reptiles are cold-blooded. Parrots are not cold-blooded. Sue’s pet parrot is not a reptile. SOLUTION a. Inductive reasoning, because a pattern is used to reach the conclusion. b. Deductive reasoning, because facts about animals and the laws of logic are used to reach the conclusion. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. If 90° < m∠R < 180°, then ∠R is obtuse. The measure of ∠R is 155°. Using the Law of Detachment, what statement can you make? 9. Use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements. If you get an A on your math test, then you can go to the movies. If you go to the movies, then you can watch your favorite actor. 10. Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show that the conjecture is true. 11. Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. All multiples of 8 are divisible by 4. 64 is a multiple of 8. So, 64 is divisible by 4. MAKING SENSE OF PROBLEMS In geometry, you will frequently use inductive reasoning to make conjectures. You will also use deductive reasoning to show that conjectures are true or false. You will need to know which type of reasoning to use. hs_geo_pe_0202.indd 79 1/19/15 9:01 AM 80 Chapter 2 Reasoning and Proofs Exercises2.2 Dynamic Solutions available at BigIdeasMath.com 1. VOCABULARY How does the prefi x “counter-” help you understand the term counterexample? 2. WRITING Explain the difference between inductive reasoning and deductive reasoning. Vocabulary and Core Concept Check In Exercises 3–8, describe the pattern. Then write or draw the next two numbers, letters, or fi gures. (See Example 1.) 3. 1, −2, 3, −4, 5, . . . 4. 0, 2, 6, 12, 20, . . . 5. Z, Y, X, W, V, . . . 6. J, F, M, A, M, . . . 7. 8. In Exercises 9–12, make and test a conjecture about the given quantity. (See Example 2.) 9. the product of any two even integers 10. the sum of an even integer and an odd integer 11. the quotient of a number and its reciprocal 12. the quotient of two negative integers In Exercises 13–16, fi nd a counterexample to show that the conjecture is false. (See Example 3.) 13. The product of two positive numbers is always greater than either number. 14. If n is a nonzero integer, then n + 1 — n is always greater than 1. 15. If two angles are supplements of each other, then one of the angles must be acute. 16. A line s divides — MN into two line segments. So, the line s is a segment bisector of — MN . In Exercises 17–20, use the Law of Detachment to determine what you can conclude from the given information, if possible. (See Example 4.) 17. If you pass the fi nal, then you pass the class. You passed the fi nal. 18. If your parents let you borrow the car, then you will go to the movies with your friend. You will go to the movies with your friend. 19. If a quadrilateral is a square, then it has four right angles. Quadrilateral QRST has four right angles. 20. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point P divides — LH into two congruent line segments. In Exercises 21–24, use the Law of Syllogism to write a new conditional statement that follows from the pair of true statements, if possible. (See Example 5.) 21. If x < −2, then ∣ x ∣ > 2. If x > 2, then ∣ x ∣ > 2. 22. If a = 3, then 5a = 15. If 1 —2 a = 1 1 —2, then a = 3. 23. If a fi gure is a rhombus, then the fi gure is a parallelogram. If a fi gure is a parallelogram, then the fi gure has two pairs of opposite sides that are parallel. 24. If a fi gure is a square, then the fi gure has four congruent sides. If a fi gure is a square, then the fi gure has four right angles. In Exercises 25–28, state the law of logic that is illustrated. 25. If you do your homework, then you can watch TV. If you watch TV, then you can watch your favorite show. If you do your homework, then you can watch your favorite show. Monitoring Progress and Modeling with Mathematics hs_geo_pe_0202.indd 80 1/19/15 9:01 AM Section 2.2 Inductive and Deductive Reasoning 81 26. If you miss practice the day before a game, then you will not be a starting player in the game. You miss practice on Tuesday. You will not start the game Wednesday. 27. If x > 12, then x + 9 > 20. The value of x is 14. So, x + 9 > 20. 28. If ∠1 and ∠2 are vertical angles, then ∠1 ≅ ∠2. If ∠1 ≅ ∠2, then m∠1 = m∠2. If ∠1 and ∠2 are vertical angles, then m∠1 = m∠2. In Exercises 29 and 30, use inductive reasoning to make a conjecture about the given quantity. Then use deductive reasoning to show that the conjecture is true. (See Example 6.) 29. the sum of two odd integers 30. the product of two odd integers In Exercises 31–34, decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. (See Example 7.) 31. Each time your mom goes to the store, she buys milk. So, the next time your mom goes to the store, she will buy milk. 32. Rational numbers can be written as fractions. Irrational numbers cannot be written as fractions. So, 1—2 is a rational number. 33. All men are mortal. Mozart is a man, so Mozart is mortal. 34. Each time you clean your room, you are allowed to go out with your friends. So, the next time you clean your room, you will be allowed to go out with your friends. ERROR ANALYSIS In Exercises 35 and 36, describe and correct the error in interpreting the statement. 35. If a fi gure is a rectangle, then the fi gure has four sides. A trapezoid has four sides. Using the Law of Detachment, you can conclude that a trapezoid is a rectangle.✗ 36. Each day, you get to school before your friend. Using deductive reasoning, you can conclude that you will arrive at school before your friend tomorrow. ✗ 37. REASONING The table shows the average weights of several subspecies of tigers. What conjecture can you make about the relation between the weights of female tigers and the weights of male tigers? Explain your reasoning. Weight of female (pounds) Weight of male (pounds) Amur 370 660 Bengal 300 480 South China 240 330 Sumatran 200 270 Indo-Chinese 250 400 38. HOW DO YOU SEE IT? Determine whether you can make each conjecture from the graph. Explain your reasoning. U.S. High School Girls’ Lacrosse N um be r of p ar ti ci pa nt s (t ho us an ds ) 20 60 100 140 x y Year 3 4 5 6 721 a. More girls will participate in high school lacrosse in Year 8 than those who participated in Year 7. b. The number of girls participating in high school lacrosse will exceed the number of boys participating in high school lacrosse in Year 9. 39. MATHEMATICAL CONNECTIONS Use inductive reasoning to write a formula for the sum of the fi rst n positive even integers. 40. FINDING A PATTERN The following are the fi rst nine Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . a. Make a conjecture about each of the Fibonacci numbers after the fi rst two. b. Write the next three numbers in the pattern. c. Research to fi nd a real-world example of this pattern. hs_geo_pe_0202.indd 81 1/19/15 9:01 AM