Chapter 2: Reasoning and Proof Why is it important to be able to think logically?

2.1 Inductive Reasoning & Conjectures Then: You used data to find patterns and make predictions Now: 1. Make conjectures based on inductive reasoning. 2. Find counterexamples.

2.1 Vocabulary Inductive reasoning: the reasoning that used information from different examples to form a conclusion. Conjecture: the concluding statement reached using inductive reasoning; educated guess.

Example 1: Patterns and Conjectures Write a conjecture that describes the pattern in the sequence. Then use the conjecture to find the next item in the sequence. a. 1, 3, 9, 27, 81 Step1: Look for a pattern: Step 2: Make a conjecture:

Example 1 b. -5, 10, -20, 40 Step1: Look for a pattern: Step 2: Make a conjecture:

Example 1: c. Step1: Look for a pattern: Step 2: Make a conjecture:

Example 2: Algebraic and Geometric Conjectures a. Make a conjecture about the sum of an odd number and an even number. Step 1: List some examples: Step 2: Look for a pattern: Step 3: Make a conjecture:

Example 2 b. Write a conjecture about the geometric relationship. 1 and  2 form a right angle. Step 1: Look for a pattern: Step 2: Make a conjecture:

Example 2 c. Write a conjecture about the geometric relationship. ABC and  DBE are vertical angles. Step 1: Look for a pattern: Step 2: Make a conjecture:

Example 3: Make Conjectures from Data The table shows the total sales for the three months a store is open. The owner wants to predict the sales for the fourth month. a. Make a statistical graph that best displays the data.

Example 3 b. Make a conjecture about the sales in the fourth month and justify your claim or prediction.

2.1 Vocabulary Counterexample: an example used to show that a given statement is not always true; can be a number, a drawing, or a statement. To show a conjecture is true for all cases, you must prove it. It only takes one false example to show that a conjecture is false.

Example 4: Find Counterexamples Determine if each conjecture is True or False. Give a counterexample for any false conjectures. a. If points A, B and C are collinear, then AB + BC = AC. b. If DEEF, then DEF is a right angle.

Example 4 c. If R and S are supplementary, and R and T are supplementary, then T and S are congruent. d. If ABC and DEF are supplementary, then ABC and DEF form a linear pair.

2.1 Assignment: Pages 95-98 #14, 18, 22-36 evens, 39 (graph a), 40-44 evens, 55, 59, 68