Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Geometry
Warm­Up
Wednesday, Jan 18
What can you logically conclude from the given statements?
1) If a student is having difficulty in class, then that student's teacher is concerned.
Elena is having difficulty in history class.
2) If a person has a job, then that person is earning money.
If a person is earning money, then that person can save money each week.
3) Solve the following equation. Justify each step.
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
TEACHER'S NOTES
Hey Guys,
Great class today. This section 2.4 is a little better than the other stuff because
you have solved similar equations before. The only new thing here is justifying
our steps, which we may have never thought about before. Study the properties
because these will allow us to make good proofs. 2.3 is like the brainteaser
questions or games.
Keep in mind the schedule for the rest of the week. Check the calendar for a
look at the rest of this week.
If you have any questions, let's get them answered early. We will review on
Friday for the Chapter 2 test on Monday. Work on your chapter 2 study guides. I
will not post anything new today. We will cover 2.5 tomorrow.
Chapter 1 Test: There are 2 people who have not taken the test. They will test
tomorrow, and I will return the grades. I decided not to enter your Chpt 1
scores yet. We have just 2 people who haven't taken the test. I will show the
scores in class tomorrow.
Mr. Sanders
PS. If you're reading this, write your name on a half­sheet of paper tomorrow
and write "I NEED 5 POINTS ON THE CHAPTER 1 EXAM." Bring the half­sheet
to my desk before the bell rings.
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Section 2.3: Deductive Reasoning
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Section 2.3: Deductive Reasoning
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Deductive Reasoning
Section 2.3
Learning Objective
To use the Law of Detachment (LoD)
To use the Law of Syllogism (LoS)
SG #9, 13, 22
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
REASONING
INDUCTIVE
REASONING
DEDUCTIVE
REASONING
Observing
Patterns
Given Statements
Conjectures
Conclusion
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Example of Deductive Reasoning
All dogs are mammals. All mammals have kidneys. Therefore, all dogs must have kidneys.
What logical conclusions can be drawn from the following statements?
1) Line l and line m are perpendicular. If two lines are perpendicular, they intersect to form right angles.
2) If two angles are supplementary, then the sum of their measures is 180o. and are supplementary.
3) If the weather is wet, then the Huskies will not play soccer. If the Huskies do not play soccer, then Nathan can stop at the ice cream shop. We will look at each of these individually...
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Law of Detachment (LoD)
If a conditional is true and its hypothesis is true, then its conclusion is true.
OR
If is a true statement and p is true, then q is true.
1) Line l and line m are perpendicular. If two lines are perpendicular, they intersect to form right angles.
True hypothesis
True conditional
Hence, the conclusion must be true: Lines l and m must intersect to form right angles.
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Law of Detachment (LoD)
If a conditional is true and its hypothesis is true, then its conclusion is true.
OR
If is a true statement and p is true, the q is true.
2) . and are supplementary. If two angles are supplementary, then the sum of their measures is 180o.
True hypothesis
True conditional
Hence, the conclusion must be true:
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Law of Syllogism (LoS)
If and are true statements, then is a true statement.
Note that q serves as the conclusion of the first conditional and the hypothesis of the second conditional. THIS HAS TO BE TRUE.
3) If the weather is wet, then the Huskies will not play soccer. If the Huskies do not play soccer, then Nathan can stop at the ice cream shop.
Hence, must be true:
If the weather is wet, Nathan can stop at the ice cream shop.
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Example
Use the LoD and LoS to draw conclusions.
The Volga River is in Europe.
If a river is less than 2300 mi long, it is not one of the world's ten longest rivers.
If a river is in Europe, then it is less than 2300 mi long. What can you logically conclude from the above statements? The Volga River is less than 2300 miles long.
The Volga River is not one of the world's ten longest rivers.
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
Homework/Classwork
Pg. 85 #5,6,8,9,10­13,15,16­21,23,25,29,32,33
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Section 2.3: Deductive Reasoning
Wednesday, Jan 18
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