Section 2.3 – Deductive Reasoning Geometry Section 2.3 – Deductive Reasoning

Note-Taking Guide I suggest only writing down things written in red

Inductive vs. Deductive Reasoning What was our definition of inductive reasoning (section 2.1)? Process of looking for patterns, making a conjecture, and then attempting to prove the conjecture true or false What is deductive reasoning? Using facts (postulates and theorems), definitions, and properties in a logical order to prove a fact

Inductive vs. Deductive Reasoning Determine if each example shows inductive or deductive reasoning Every junior Joe knows is older than every sophomore he knows. Therefore Joe reasons that Bill, a junior, is older than Sam, a sophomore. Inductive He uses the pattern of juniors being older to make a conjecture that Bill is older than Sam) Joe knows that Greg is older than Sam. Joe knows that Bill is older than Greg. Therefore Joe reasons that Bill is older than Sam. Deductive He uses two facts in a logical order to prove Bill is older than Sam

Two Laws of Deductive Reasoning First some symbols: Lowercase letters, like “p” stand for either a hypothesis or conclusion A tilde like “ ~ ” stands for negation An arrow like “→” stands for “if-then” A double arrow like “↔” stands for “if and only if” Example: If an angle measures 130°, then the angle is obtuse. Let p = “an angle measures 130°“ Let q = “the angle is obtuse” This statement can be represented by 𝑝→𝑞 which you say as “if p, then q” The inverse would be written as ~𝑝→~𝑞

Two Laws of Deductive Reasoning Law of Detachment If 𝑝→𝑞 is true and you are given a specific case when 𝑝 is true, then you know 𝑞 is true Note: If you have a specific case when 𝑞 is true, you do NOT know that 𝑝 is true

Two Laws of Deductive Reasoning Example 1 If an angle measures 23°, then it is acute. Given 𝑚∠𝐴𝐵𝐶=23°, make a valid conclusion Answer: ∠𝐴𝐵𝐶 is acute Example 2 If an angle measures 152°, then it is obtuse. Given ∠𝐷𝐸𝐹 is obtuse, make a valid conclusion Answer: no valid conclusion We do not know that ∠𝐷𝐸𝐹 must be 152°, so we cannot make any valid conclusion

Two Laws of Deductive Reasoning Law of Syllogism If 𝑝→𝑞 and 𝑞→𝑟 are true statements, then 𝑝→𝑟 is a true statement Example 1: Joe is taller than Fred. Fred is taller than Bill. Make a valid statement using Syllogism Answer: Joe is taller than Bill

Two Laws of Deductive Reasoning Example 2 – Make a valid statement using Syllogism If a smoke detector makes a loud beep, students will leave the building. If a smoke detector senses high levels of smoke and heat, then it will make a loud beep. If there is smoke and heat, then a smoke detector will detect high levels of smoke and heat. If there is a fire, then there will be smoke and heat. Answer: If there is a fire, then students will leave the building