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2 2 Lesson What You Will Learn,Use inductive reasoning. Use deductive reasoning,Core Vocabul,Vocabulary,conjecture p 76 Using Inductive Reasoning. inductive reasoning p 76,counterexample p 77,deductive reasoning p 78. Core Concept,Inductive Reasoning, A conjecture is an unproven statement that is based on observations You use. inductive reasoning when you find a pattern in specific cases and then write a. conjecture for the general case,Describing a Visual Pattern.
Describe how to sketch the fourth figure in the pattern Then sketch the fourth figure. Figure 1 Figure 2 Figure 3, Each circle is divided into twice as many equal regions as the figure number Sketch. the fourth figure by dividing a circle into eighths Shade the section just above the. horizontal segment at the left, Monitoring Progress Help in English and Spanish at BigIdeasMath com. 1 Sketch the fifth figure in the pattern in Example 1. Sketch the next figure in the pattern,76 Chapter 2 Reasoning and Proofs. Making and Testing a Conjecture, Numbers such as 3 4 and 5 are called consecutive integers Make and test a. conjecture about the sum of any three consecutive integers. Step 1 Find a pattern using a few groups of small numbers. 3 4 5 12 4 3 7 8 9 24 8 3,10 11 12 33 11 3 16 17 18 51 17 3.
Step 2 Make a conjecture, Conjecture The sum of any three consecutive integers is three times the. second number, Step 3 Test your conjecture using other numbers For example test that it works with. the groups 1 0 1 and 100 101 102,1 0 1 0 0 3,100 101 102 303 101 3. Core Concept,Counterexample, To show that a conjecture is true you must show that it is true for all cases You. can show that a conjecture is false however by finding just one counterexample. A counterexample is a specific case for which the conjecture is false. Finding a Counterexample, A student makes the following conjecture about the sum of two numbers Find a.
counterexample to disprove the student s conjecture. Conjecture The sum of two numbers is always more than the greater number. To find a counterexample you need to find a sum that is less than the greater number. Because a counterexample exists the conjecture is false. Monitoring Progress Help in English and Spanish at BigIdeasMath com. 4 Make and test a conjecture about the sign of the product of any three. negative integers, 5 Make and test a conjecture about the sum of any five consecutive integers. Find a counterexample to show that the conjecture is false. 6 The value of x2 is always greater than the value of x. 7 The sum of two numbers is always greater than their difference. Section 2 2 Inductive and Deductive Reasoning 77,Using Deductive Reasoning. Core Concept,Deductive Reasoning, Deductive reasoning uses facts definitions accepted properties and the laws of. logic to form a logical argument This is different from inductive reasoning which. uses specific examples and patterns to form a conjecture. Laws of Logic,Law of Detachment, If the hypothesis of a true conditional statement is true then the conclusion is. Law of Syllogism,If hypothesis p then conclusion q.
If these statements are true,If hypothesis q then conclusion r. If hypothesis p then conclusion r then this statement is true. Using the Law of Detachment, If two segments have the same length then they are congruent You know that. BC XY Using the Law of Detachment what statement can you make. Because BC XY satisfies the hypothesis of a true conditional statement the. conclusion is also true,Using the Law of Syllogism. If possible use the Law of Syllogism to write a new conditional statement that follows. from the pair of true statements,a If x2 25 then x2 20. If x 5 then x2 25, b If a polygon is regular then all angles in the interior of the polygon are congruent.
If a polygon is regular then all its sides are congruent. a Notice that the conclusion of the second statement is the hypothesis of the first. statement The order in which the statements are given does not affect whether you. can use the Law of Syllogism So you can write the following new statement. If x 5 then x2 20, b Neither statement s conclusion is the same as the other statement s hypothesis. You cannot use the Law of Syllogism to write a new conditional statement. 78 Chapter 2 Reasoning and Proofs,Using Inductive and Deductive Reasoning. What conclusion can you make about the product of an even integer and any. other integer, Step 1 Look for a pattern in several examples Use inductive reasoning to make a. conjecture,MAKING SENSE,OF PROBLEMS 2 2 4 1 2 2 2 2 4 3 2 6. In geometry you will 2 4 8 1 4 4 2 4 8 3 4 12,frequently use inductive.
reasoning to make Conjecture Even integer Any integer Even integer. conjectures You will also Step 2 Let n and m each be any integer Use deductive reasoning to show that the. use deductive reasoning conjecture is true,to show that conjectures. are true or false You will 2n is an even integer because any integer multiplied by 2 is even. need to know which type 2nm represents the product of an even integer 2n and any integer m. of reasoning to use, 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, 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,Section 2 2 Inductive and Deductive Reasoning 79.
2 2 Exercises Dynamic Solutions available at BigIdeasMath com. Vocabulary and Core Concept Check, 1 VOCABULARY How does the prefix counter help you understand the term counterexample. 2 WRITING Explain the difference between inductive reasoning and deductive reasoning. Monitoring Progress and Modeling with Mathematics, In Exercises 3 8 describe the pattern Then write or In Exercises 17 20 use the Law of Detachment to. draw the next two numbers letters or figures determine what you can conclude from the given. See Example 1 information if possible See Example 4. 3 1 2 3 4 5 4 0 2 6 12 20 17 If you pass the final then you pass the class You. passed the final,5 Z Y X W V 6 J F M A M, 18 If your parents let you borrow the car then you will. 7 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. 8 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. into two congruent line segments,divides LH, In Exercises 9 12 make and test a conjecture about the. In Exercises 21 24 use the Law of Syllogism to write a. given quantity See Example 2, new conditional statement that follows from the pair of. 9 the product of any two even integers true statements if possible See Example 5. 21 If x 2 then x 2 If x 2 then x 2,10 the sum of an even integer and an odd integer. 22 If a 3 then 5a 15 If 2 a 1 2 then a 3,11 the quotient of a number and its reciprocal. 23 If a figure is a rhombus then the figure is a,12 the quotient of two negative integers.
parallelogram If a figure is a parallelogram then,the figure has two pairs of opposite sides that. In Exercises 13 16 find a counterexample to show that. are parallel,the conjecture is false See Example 3. 13 The product of two positive numbers is always greater 24 If a figure is a square then the figure has four. than either number congruent sides If a figure is a square then the. figure has four right angles, 14 If n is a nonzero integer then is always greater. n In Exercises 25 28 state the law of logic that,is illustrated. 15 If two angles are supplements of each other then one. 25 If you do your homework then you can watch TV If. of the angles must be acute, you watch TV then you can watch your favorite show.
into two line segments So the, A line s divides MN If you do your homework then you can watch your. line s is a segment bisector of MN favorite show,80 Chapter 2 Reasoning and Proofs. 26 If you miss practice the day before a game then you 37 REASONING The table shows the average weights. will not be a starting player in the game of several subspecies of tigers What conjecture can. you make about the relation between the weights of. You miss practice on Tuesday You will not start the. female tigers and the weights of male tigers Explain. game Wednesday,your reasoning,27 If x 12 then x 9 20 The value of x is 14. So x 9 20 Weight Weight,of female of male, 28 If 1 and 2 are vertical angles then 1 2 pounds pounds. If 1 2 then m 1 m 2 Amur 370 660, If 1 and 2 are vertical angles then m 1 m 2 Bengal 300 480.
In Exercises 29 and 30 use inductive reasoning to South China 240 330. make a conjecture about the given quantity Then use Sumatran 200 270. deductive reasoning to show that the conjecture is true. See Example 6 Indo Chinese 250 400,29 the sum of two odd integers. 30 the product of two odd integers 38 HOW DO YOU SEE IT Determine whether you. can make each conjecture from the graph Explain, In Exercises 31 34 decide whether inductive reasoning your reasoning. or deductive reasoning is used to reach the conclusion. Explain your reasoning See Example 7 U S High School Girls Lacrosse. Number of participants, 31 Each time your mom goes to the store she buys milk y. So the next time your mom goes to the store she will. buy milk thousands 100, 32 Rational numbers can be written as fractions 60. Irrational numbers cannot be written as fractions,So 12 is a rational number.
1 2 3 4 5 6 7x, 33 All men are mortal Mozart is a man so Mozart Year. 34 Each time you clean your room you are allowed to a More girls will participate in high school lacrosse. go out with your friends So the next time you clean in Year 8 than those who participated in Year 7. your room you will be allowed to go out with b The number of girls participating in high. your friends school lacrosse will exceed the number of boys. participating in high school lacrosse in Year 9, ERROR ANALYSIS In Exercises 35 and 36 describe and. correct the error in interpreting the statement, 35 If a figure is a rectangle then the figure has four sides 39 MATHEMATICAL CONNECTIONS Use inductive. A trapezoid has four sides reasoning to write a formula for the sum of the. first n positive even integers,Using the Law of Detachment you can. conclude that a trapezoid is a rectangle, 40 FINDING A PATTERN The following are the first nine.
Fibonacci numbers,1 1 2 3 5 8 13 21 34, 36 Each day you get to school before your friend a Make a conjecture about each of the Fibonacci. numbers after the first two,Using deductive reasoning you can. conclude that you will arrive at school,before your friend tomorrow. b Write the next three numbers in the pattern,c Research to find a real world example of. this pattern,Section 2 2 Inductive and Deductive Reasoning 81.
41 MAKING AN ARGUMENT Which argument is correct 45 DRAWING CONCLUSIONS Decide whether each. Explain your reasoning conclusion is valid Explain your reasoning. Argument 1 If two angles measure 30 and 60 Yellowstone is a national park in Wyoming. then the angles are complementary 1 and 2 are You and your friend went camping at. complementary So m 1 30 and m 2 60 Yellowstone National Park. Argument 2 If two angles measure 30 and 60 then When you go camping you go canoeing. the angles are complementary The measure of 1 is,If you go on a hike your friend goes with you. 30 and the measure of 2 is 60 So 1 and 2,are complementary You go on a hike. There is a 3 mile long trail near your campsite, 42 THOUGHT PROVOKING The first two terms of a a You went camping in Wyoming. sequence are 14 and 12 Describe three different possible. patterns for the sequence List the first five terms for b Your friend went canoeing. each sequence c Your friend went on a hike,d You and your friend went on a hike on a. 43 MATHEMATICAL CONNECTIONS Use the table to 3 mile long trail. make a conjecture about the relationship between,x and y Then write an equation for .
Section 2 2 Inductive and Deductive Reasoning 77 Making and Testing a Conjecture Numbers such as 3 4 and 5 are called consecutive integers Make and test a conjecture about the sum of any three consecutive integers