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

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

3. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Warm Up 1. January, March, May,... The next month is July. Alternating months of the year make up the pattern. 2. 7, 14, 21, 28, … The next multiple is 35. Multiples of 7 make up the pattern. In this pattern, the figure rotates 90° counter-clockwise each time. The next figure is. 3.

4. Holt McDougal 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.

5. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Complete the conjecture. Example 1A: Making a Conjecture The sum of two positive numbers is ?. The sum of two positive numbers is positive. List some examples and look for a pattern. 1 + 1 = 23.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917

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

7. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Check It Out! Example 1 The product of two odd numbers is ?. Complete the conjecture. The product of two odd numbers is odd. List some examples and look for a pattern. 1  1 = 1 3  3 = 9 5  7 = 35

8. Holt McDougal 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.

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

10. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 2A: Finding a Counterexample For every integer n, n 3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n 3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n 3 = –27 and –27  0, the conjecture is false. n = –3 is a counterexample.

11. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 2B: Finding a Counterexample Two complementary angles are not congruent. If the two congruent angles both measure 45°, the conjecture is false. 45° + 45° = 90°

12. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Check It Out! Example 2 For any real number x, x 2 ≥ x. Show that the conjecture is false by finding a counterexample. Let x =. 1212 The conjecture is false. Since =, ≥. 1212 2 1212 1414 1414

13. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Directions Test your skills at proving conjectures TRUE or FALSE with a counterexample. With your partner - Go around the room and read the various conjectures: 1. If you think its FALSE, write an counterexample to prove it false on the paper below and put names by counterexample. 2. If you think its TRUE, write TRUE on paper below and put names by it. One point will be awarded to each unique counterexample or correct TRUE statement. One point will be deducted for each incorrect statement.

14. Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Determine if each conjecture is true. If false, give a counterexample. A. The quotient of two negative numbers is a positive number. B. Every prime number is odd. C. Two supplementary angles are not congruent. D. The square of an odd integer is odd. E. Three points on a plane always form a triangle F. If y-1>0, then 0<y<1 false; 2 true false; 90° and 90° true