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LOGIC, REASONING, & PROOF CH 1-1 USING PATTERNS & INDUCTIVE REASONING Inductive reasoning is a type of reasoning that allows you to reach conclusions based on pattern of specific examples or past events. Example 1 a. Find the next two terms in this sequence: 2, 4, 6, 8,… Describe the pattern. b. Find the next two terms in this sequence: 3, 6, 12, 24,… Describe the pattern. c. Find the next two terms in this sequence: 1, 2, 4, 5, 10, 11, 22,… Describe the pattern. A conclusion reached by using inductive reasoning is sometimes called a conjecture. Example 2 Use inductive reasoning to find the sum of the first 20 odd numbers. Ch 1-7 Using Deductive Reasoning Deductive reasoning is a process of reasoning logically from given facts to a conclusion. Properties of congruence: Reflexive property: Any geometric object is congruent to itself. Symmetric property: If one geometric object is congruent to a second, then the second is congruent to the first. Transitive property: If one geometric object is congruent to a second, and the second is congruent to a third, then the first object is congruent to the third object. Properties of equality Addition Property Subtraction Property Multiplication Property Division Property Substitution Property Distributive Property Example 1 Give a reason for each step 4x – 6 = 94 4x = 100 x = 25 A convincing argument that uses deductive reasoning is also called a proof. A conjecture that is proven is a theorem. Theorems Vertical angles theorem: Vertical angles are congruent. Congruent Supplements Theorem: If two angles are supplements of congruent angles (or the same angle), then the two angles are congruent. Congruent Complements Theorem: If two angles are complements of congruent angles (or the same angle), then the two angles are congruent. Example 2 Use a 2-Column proof (Statement & Reason) to prove the Congruent Supplements Theorem. Ch 4-1 Using Logical Reasoning Another name for an if-then statement is a conditional. Every conditional has two parts. The part following the if is the hypothesis, and the part following the then is the conclusion. Example 1 Identify the hypothesis and the conclusion in this statement. If it is February, then there are only 28 days in the month. When you determine whether a conditional is true or false, you determine its truth value. To show that a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false. Example 2 Is the conditional from example 1 true or false? Explain. Example 3 Find a counterexample for this conditional: If the name of a state contains the word New, then the state border ocean. The converse of a conditional interchanges the hypothesis and conclusions. Example 4 Write the converse of this statement: If a polygon is a quadrilateral, then it has four sides. When a conditional and its converse are true, you can combine them as biconditional (if and only if). Example 5 Write a biconditional statement for the example 4 problem. Example 6 Write the two statements that make up this definition: A right angle has a measure of 90o. Then write the definition as a biconditional. Conditional: Converse: Biconditional: The negation of a statement has the opposite meaning. Example 7 Write the negation of each statement. a.) An angle is acute b.) two lines are not parallel The inverse of a conditional negates both the hypothesis and the conclusion. The contrapositive of a conditional interchanges and negates both the hypothesis and the conclusion. Example 8 a. Write the following for this statement: If it is raining, then it is cloudy. Conditional: Converse: Inverse: Contrapositive: Biconditional: b. What are the truth values for each statement. Conditional: Converse: Inverse: Contrapositive: Biconditional: