8 CHAPTER 1 Critical Thinking Skills 44. Prove, using deductive reasoning, the conjecture you made in Exercise 40 part c). + + + = + + + = n n n n n n , 5, ( 5) 2 5,2 511 + + = + + − = n n n n n 2 16, 2 16 2 8, 8 8 In Exercises 45–50, determine a counterexample to show that each statement is incorrect. 45. The product of any two counting numbers is divisible by 2. 3 5 15, × = which is not divisible by 2. 46. The sum of any three two-digit numbers is a three-digit number. 10 11 12 33, + + = which is not a three-digit number. 47. When a counting number is added to 3 and the sum is divided by 2, the quotient will be an even number. (3 2)/2 5/2, + = which is not an even number. 48. The product of any two three-digit numbers is a five-digit number. 900 900 810,000, × = which is not a five-digit number. 49. The difference of any two counting numbers will be a counting number. 1 2 1, − = − which is not a counting number. 50. The sum of any two odd numbers is divisible by 4. 1 5 6, + = which is not divisible by 4. 51. Interior Angles of a Triangle a) Construct a triangle and measure the three interior angles with a protractor. What is the sum of the measures? The sum of the measures of the interior angles should be 180°. b) Construct three other triangles, measure the angles, and record the sums. Are your answers the same? Yes, the sum of the measures of the interior angles should be 180°. c) Make a conjecture about the sum of the measures of the three interior angles of a triangle. The sum of the measures of the interior angles of a triangle is 180°. 52. Interior Angles of a Quadrilateral a) Construct a quadrilateral (a four-sided figure) and measure the four interior angles with a protractor. What is the sum of the measures? The sum of the measures of the interior angles should be 360°. b) Construct three other quadrilaterals, measure the angles, and record the sums. Are your answers the same? Yes, the sum of the measures of the interior angles should be 360°. c) Make a conjecture about the sum of the measures of the four interior angles of a quadrilateral. The sum of the measures of the interior angles of a quadrilateral is 360°. Concept/Writing Exercises 53. Computer Log On While logging on to your computer, you type in your username followed by what you believe is your password. The computer indicates that a mistake has been made and asks you to try again. You retype your username and the same password. Again, the computer indicates a mistake has been made. You decide not to try again, reasoning you will get the same error message from the computer. What type of reasoning did you use? Explain. Inductive reasoning; a general conclusion is obtained from observation of specific cases. 37. Pick any number, multiply the number by 3, add 6 to the product, divide the sum by 3, and subtract 2 from the quotient. See Example 5. a) What is the relationship between the number you started with and the final number? You should obtain the original number. b) Arbitrarily select some different numbers and repeat the process, recording the original number and the result. You should obtain the original number. c) Make a conjecture about the relationship between the original number and the final number. The result is the original number. 38. Pick any number and multiply the number by 4. Add 6 to the product. Divide the sum by 2 and subtract 3 from the quotient. a) What is the relationship between the number you started with and the final answer? You should obtain twice the original number. b) Arbitrarily select some different numbers and repeat the process, recording the original number and the results. You should obtain twice the original number. c) Make a conjecture about the relationship between the original number and the final number. The result is always twice the original number. 39. Pick any number and add 1 to it. Determine the sum of the new number and the original number. Add 9 to the sum. Divide the new sum by 2 and subtract the original number from the quotient. a) What is the final number? 5 b) Arbitrarily select some different numbers and repeat the process. Record the results. You should obtain the number 5. c) Make a conjecture about the final number. The result is always the number 5. 40. Pick any number and add 5 to it. Determine the sum of the new number and the original number. Add 11 to the sum. Divide the new sum by 2 and subtract the original number from the quotient. a) What is the result? 8 b) Arbitrarily select some different numbers and repeat the process. Record the results. You should obtain the number 8. c) Make a conjecture about the final number. The result is always the number 8. 41. Prove, using deductive reasoning, the conjecture you made in Exercise 37 part c). + + = + + − = n n n n n n n , 3 , 3 6, 3 6 3 2, 2 2 42. Prove, using deductive reasoning, the conjecture you made in Exercise 38 part c). n n n n n n n , 4 , 4 6, 4 6 2 2 3, 2 3 3 2 + + = + + − = 43. Prove, using deductive reasoning, the conjecture you made in Exercise 39 part c). n n n n n n n , 1, ( 1)2 1,2 192 10, + + + = + + + = + + = + + − = n n n n 2 10 2 5, 5 5
RkJQdWJsaXNoZXIy NjM5ODQ=