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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures 2-1 Using Inductive Reasoning to Make Conjectures Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Are you ready? Complete each sentence. 1. ? points are points that lie on the same line. 2. ? points are points that lie in the same plane. 3. The sum of the measures of two ? angles is 90°.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures TSW use inductive reasoning to identify patterns and make conjectures. TSW find counterexamples to disprove conjectures. Objectives

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Biologists use inductive reasoning to develop theories about migration patterns.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures inductive reasoning conjecture counterexample Vocabulary

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 1: Identifying a Pattern January, March, May,...

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 2: Identifying a Pattern 7, 14, 21, 28, …

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 3: Identifying a Pattern

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 4 Find the next item in the pattern 0.4, 0.04, 0.004, …

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Complete the conjecture. Example 5: Making a Conjecture The sum of two positive numbers is ?.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Complete the conjecture. Example 6: Making a Conjecture The number of lines formed by 4 points, no three of which are collinear, is ?.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 7 The product of two odd numbers is ?. Complete the conjecture.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 8: Biology Application The cloud of water leaving a whales blowhole when it exhales is called its blow. A biologist observed blue-whale blows of 25 ft, 29 ft, 27 ft, and 24 ft. Another biologist recorded humpback- whale blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a conjecture based on the data. Heights of Whale Blows Height of Blue-whale Blows Height of Humpback-whale Blows 8789

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 9 Make a conjecture about the lengths of male and female whales based on the data. Average Whale Lengths Length of Female (ft) Length of Male (ft)

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. To show that a conjecture is always true, you must prove it. A counterexample can be a drawing, a statement, or a number.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 10: Finding a Counterexample For every integer n, n 3 is positive.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 11: Finding a Counterexample Two complementary angles are not congruent.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 12: Finding a Counterexample The monthly high temperature in Abilene is never below 90°F for two months in a row. Monthly High Temperatures (ºF) in Abilene, Texas JanFebMarAprMayJunJulAugSepOctNovDec

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 13 For any real number x, x 2 x. Show that the conjecture is false by finding a counterexample.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 14 Supplementary angles are adjacent. Show that the conjecture is false by finding a counterexample.

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 15 The radius of every planet in the solar system is less than 50,000 km. Show that the conjecture is false by finding a counterexample. Planets Diameters (km) MercuryVenusEarthMarsJupiterSaturnUranusNeptunePluto ,10012, ,000121,00051,10049,

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures

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Holt Geometry 2-1 Using Inductive Reasoning to Make Conjectures Lesson Quiz Find the next item in each pattern , 0.07, 0.007, … Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. 4. Every prime number is odd. 5. Two supplementary angles are not congruent. 6. The square of an odd integer is odd. false; 2 true false; 90° and 90° true

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