Using Inductive Reasoning to 2 1 Make Conjectures

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

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

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

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

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

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

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

Using Inductive Reasoning to 2 -1 Make Conjectures Check It Out! Example 1 Find the next item in the pattern 0. 4, 0. 004, … Holt Geometry

Using Inductive Reasoning to 2 -1 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. Holt Geometry

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

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

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

Using Inductive Reasoning to 2 -1 Make Conjectures Check It Out! Example 3 Make a conjecture about the lengths of male and female whales based on the data. Average Whale Lengths Length of Female (ft) 49 51 50 48 51 47 Length of Male (ft) 47 45 44 46 48 48 Holt Geometry

Using Inductive Reasoning to 2 -1 Make Conjectures To show that a conjecture is always true, you must prove it. 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. A counterexample can be a drawing, a statement, or a number. Holt Geometry

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

Using Inductive Reasoning to 2 -1 Make Conjectures Example 4 A: Finding a Counterexample Show that the conjecture is false by finding a counterexample. For every integer n, n 3 is positive. Holt Geometry

Using Inductive Reasoning to 2 -1 Make Conjectures Example 4 B: Finding a Counterexample Show that the conjecture is false by finding a counterexample. Two complementary angles are not congruent. Holt Geometry

Using Inductive Reasoning to 2 -1 Make Conjectures Example 4 C: Finding a Counterexample Show that the conjecture is false by 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 Jan Feb Mar Apr May Jun Jul Aug Sep 88 89 97 99 107 109 110 107 106 103 Holt Geometry Oct Nov Dec 92 89

Using Inductive Reasoning to 2 -1 Make Conjectures Check It Out! Example 4 a Show that the conjecture is false by finding a counterexample. For any real number x, x 2 ≥ x. Holt Geometry

Using Inductive Reasoning to 2 -1 Make Conjectures Check It Out! Example 4 b Show that the conjecture is false by finding a counterexample. Supplementary angles are adjacent. Holt Geometry

Using Inductive Reasoning to 2 -1 Make Conjectures Check It Out! Example 4 c Show that the conjecture is false by finding a counterexample. The radius of every planet in the solar system is less than 50, 000 km. Planets’ Diameters (km) Mercury Venus Earth Mars 4880 12, 100 Holt Geometry 12, 800 6790 Jupiter Saturn Uranus Neptune Pluto 143, 000 121, 000 51, 100 49, 500 2300

Using Inductive Reasoning to 2 -1 Make Conjectures Lesson Quiz Find the next item in each pattern. 1. 0. 7, 0. 007, … 2. 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. Holt Geometry