1. 2.2 Continued: Deductive Reasoning
Geometry
2.2 Continued: Deductive Reasoning
Essential Question: How is deductive reasoning different than inductive reasoning?

2. 2.2 Continued Deductive Reasoning
Topic/Objectives
Use symbolic notation to represent logical statements.
Form conclusions using logic.
To use deductive reasoning.
June 11, 2018
2.2 Continued Deductive Reasoning

3. Deductive Reasoning Geometric proofs always use deductive reasoning.
Uses facts, definitions and properties in a logical order to make a logical argument.
Geometric proofs always use deductive reasoning.
June 11, 2018
2.2 Continued Deductive Reasoning

4. 2.2 Continued Deductive Reasoning
Rules of Deduction
Law of Detachment
Law of Syllogism
June 11, 2018
2.2 Continued Deductive Reasoning

5. Law of Detachment If P  Q is true, and P is true, then Q is true.
Example:
If I get an A in Geometry, then I will get a new bike.
I get an A in Geometry.
Conclude:
I get a new bike.
June 11, 2018
2.2 Continued Deductive Reasoning

6. 2.2 Continued Deductive Reasoning
Example
Judy’s parents tell her that if she wants to go to the movies Saturday night, she has to wash the windows.
Judy gets up Saturday and washes all the windows.
What do we conclude?
Judy can go to the movies that night.
June 11, 2018
2.2 Continued Deductive Reasoning

7. 2.2 Continued Deductive Reasoning
What about this?
If two angles form a linear pair, then they are supplementary.
A and B are supplementary.
Conclusion: A and B form a linear pair.
WRONG!
This is saying P Q, and Q is true. This doesn’t make P true. 1 2
1 and 2 are supp, but not a linear pair.
June 11, 2018
2.2 Continued Deductive Reasoning

8. 2.2 Continued Deductive Reasoning
Commonly seen errors.
“If there is a storm, then the power will go out”, said Marco.
Suddenly the power goes out, leaving everyone in the dark.
“Gee, I guess there is a storm!”, exclaims Marco.
“No, you dummy: you forgot to pay the electric bill”, said his sister.
Just because the conclusion is true, does not mean that the hypothesis is true.
June 11, 2018
2.2 Continued Deductive Reasoning

9. Law of Detachment (again)
P  Q is true.
P is true.
Conclusion: Q is true.
P  Q P Q
June 11, 2018
2.2 Continued Deductive Reasoning

10. 2.2 Continued Deductive Reasoning
Law of Syllogism
In symbols:
If P  Q is true, and Q  R is true, then P  R is also true.
Example:
If Sam gets an A in math, then he can buy a car.
If he can buy a car, then he will be popular.
Conclusion:
If Sam gets an A in math, then he will be popular.
June 11, 2018
2.2 Continued Deductive Reasoning

11. 2.2 Continued Deductive Reasoning
Example
Write the conclusion that we can make based on the law of syllogism.
If I watch too much TV, then I will be lazy.
If I am lazy, then I won’t get an A in math.
Conclusion:
If I watch too much TV, then I won’t get an A in math.
June 11, 2018
2.2 Continued Deductive Reasoning

12. 2.2 Continued Deductive Reasoning
Extending a Syllogism
If I wake up, then I can get dressed.
If I get dressed, then I can go to school.
If I go to school, then I can walk to class.
If I walk to class, then I will meet the school bully.
If I meet the school bully, then I will get in a fight.
If I get in a fight, then I will be suspended.
If I get suspended, then I will stay home.
If I stay home, then I will go to sleep.
Conclusion:
If I wake up, then I will go to sleep.
June 11, 2018
2.2 Continued Deductive Reasoning

13. 2.2 Continued Deductive Reasoning
So where’s the math?
If angles form a linear pair, then they are supplementary.
If angles are supplementary, then their sum is 180°.
Conclusion:
If angles form a linear pair, then their sum is 180°.
We have just proved a theorem!
June 11, 2018
2.2 Continued Deductive Reasoning

14. 2.2 Continued Deductive Reasoning
Syllogism (again)
Jump from beginning to the end.
If P  Q, and Q  R, then P  R
These are the same means…
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2.2 Continued Deductive Reasoning

15. Putting Two Laws Together
If Jake gets a job, then he can pay for football camp. If he goes to football camp, then he will make the varsity team. If he makes varsity, then he will be Homecoming King.
Jake gets a job.
Conclusion: Jake will be Homecoming King.
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2.2 Continued Deductive Reasoning

16. 2.2 Continued Deductive Reasoning
Any Questions?
June 11, 2018
2.2 Continued Deductive Reasoning

17. 2.2 Continued Deductive Reasoning
Practice
4. Determine if statement (3) follows from statements (1) and (2). If so, what Law was used?
If you eat sushi, then you will get sick.
You eat sushi.
You get sick.
Yes this follows, by Law of Detachment.
June 11, 2018
2.2 Continued Deductive Reasoning

18. 2.2 Continued Deductive Reasoning
Practice
5. Determine if statement (3) follows from statements (1) and (2). If so, what Law was used?
If x = 10, then y = 20
If y = 20, then z = 25
If x = 10, then z = 25
Yes this follows by the Law of Syllogism.
June 11, 2018
2.2 Continued Deductive Reasoning

19. 2.2 Continued Deductive Reasoning
Practice
6. Determine if statement (3) follows from statements (1) and (2). If so, what Law was used?
If you are a lawyer, then you will be rich.
You are rich.
Then you are a lawyer
No, this doesn’t follow. The argument is invalid.
You could get rich another way.
June 11, 2018
2.2 Continued Deductive Reasoning

20. 2.2 Continued Deductive Reasoning
Last Practice Q P
7. If Jim talks back to his boss, then he will lose his job. If he loses his job, he will have to sell his car. Jim talks back to the boss.
What’s the conclusion?
Jim has to sell his car. Q R P
P  Q, Q  R
means P  R
(Syllogism)
P  R
P is True
R is True
(Detachment)
June 11, 2018
2.2 Continued Deductive Reasoning