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Inductive Reasoning and Conjecture Section 2-1 Objective: Find counterexamples Conjecture An educated guess based on known information. Inductive Reasoning Reasoning that uses a number of specific examples to arrive at a conjecture. Example of Inductive Reasoning The last five mornings I drove to Auburn, the traffic was heavy on Wednesdays and light on Sundays. Conclusion: Weekdays have heavier traffic than weekends. Example 1 Finding a Pattern using Inductive Reasoning Make a conjecture about the next triangular number. Example 2 Make a conjecture about the next number. -8, -5, -2, 1, 4, ____ Practice 1 Make a conjecture about the next number in the sequence. 1. 1, 2, 4, 8, 16, ____ 2. 4, 6, 9, 13, 18, _____ 3. 1/2, 1/4, 1/8, 1/16, ____ Example 3 Make a conjecture. Draw an example to support your conjecture. 1. For points P, Q, and R, PQ = 9, QR = 15, and PR = 12. Example 3 2. < 3 and < 4 are a linear pair. Example 3 3. The sum of two odd numbers. Practice 2 1. The sum of two even numbers. 2. The relationship between AB and EF if AB = CD and CD = EF. 3. The sum of squares of two consecutive natural numbers. Counterexample One case that the conjecture does not work. Proves a conjecture wrong. Example 4 Determine whether each is true or false. Give a counterexample for a false statement. 1. If n is a real number, then n2 > n Example 4 2. If JK = KL, then K is the midpoint of JL Example 4 3. If n is a real number, then –n is a negative number. Practice 3 Determine whether each is true or false. Give a counterexample for a false statement. 1. If <ABC = <DBE, then <ABC and <DBE are vertical angles. 2. If <A and <B are supplementary, then they share a side. 3. If <1 and <2 are adjacent angles, then <1 and <2 form a linear pair.