2.2 Deductive Reasoning Objective:

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Presentation transcript:

2.2 Deductive Reasoning Objective: I CAN use inductive and deductive reasoning to make and defend conjectures. Serra - Discovering Geometry Chapter 2: Reasoning in Geometry

Vocabulary Conditional Statement: hypothesis & conclusion If-Then Form: If hypothesis, then conclusion. Ex: All math classes use numbers. If the class is math, then it uses numbers. Negation: opposite (not) Ex: Math isn’t the best class ever. Math is the best class ever. Serra - Discovering Geometry Chapter 2: Reasoning in Geometry

Vocabulary Converse: If conclusion, then hypothesis. Ex: If an angle is 90, then it’s right. If an angle is right, then it’s 90. Inverse: If NOT hypothesis, then NOT conclusion. Ex: If an angle is 90, then it’s right. If an angle is not 90, then it’s not right. Contrapositive: If NOT conclusion, then NOT hypothesis. Ex: If an angle is 90, then it’s right. If an angle is not right, then it’s not 90. Serra - Discovering Geometry Chapter 2: Reasoning in Geometry 3

Vocabulary Biconditional Statement: Perpendicular Lines: If 2 lines intersect to form a right angle, then they are perpendicular lines. If 2 lines are perpendicular, then they intersect to form a right angle. 2 lines are perpendicular if and only if they intersect to form a right angle. Biconditional Statement: statement and its converse are true “if and only if” Serra - Discovering Geometry Chapter 2: Reasoning in Geometry 4

Vocabulary Deductive Reasoning: proof using definitions, postulates, theorems Law of Detachment: If the hypothesis is true, then the conclusion is true. Law of Syllogism: If p, then q. If q, then r. If p, then r. Serra - Discovering Geometry Chapter 2: Reasoning in Geometry 5

Example #1 What conclusion can you draw based on the two conditional statements below? Law of Syllogism Serra - Discovering Geometry Chapter 2: Reasoning in Geometry

Example #2 Solve the equation for x. Give a reason for each step in the process. Original Equation Combine Like Terms Distribute Combine Like Terms Subtract 12 Add 5x Divide by 15 Serra - Discovering Geometry Chapter 2: Reasoning in Geometry 7

Example #3 Acute Acute Acute In each diagram ray AC bisects obtuse angle BAD. Classify each angle as acute, right, or obtuse. Then write and prove a conjecture about the angles formed. D A C B D C A B A B D C Acute Acute Acute Serra - Discovering Geometry Chapter 2: Reasoning in Geometry 8

If an obtuse angle is bisected, Example #3 In each diagram ray AC bisects obtuse angle BAD. Classify each angle as acute, right, or obtuse. Then write and prove a conjecture about the angles formed. Conjecture: If an obtuse angle is bisected, then the two newly formed congruent angles are _________. acute Statements Reasons 1. Given 2. Definition of Angle Bisector 3. Definition of Acute Angle Serra - Discovering Geometry Chapter 2: Reasoning in Geometry 9