1. Section 2.3 – Deductive Reasoning
Geometry
Section 2.3 – Deductive Reasoning

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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