Honors Geometry

1
Honors Geometry

• Lesson 3.3
• Using the Laws of Logic

2
What You Should LearnWhy You Should Learn It

• Goal 1 How to form the negation of a statement
  and the contrapositive of a conditional statement
• Goal 2 How to use a syllogism to reason
  deductively
• You can improve your ability to reason logically,
  both in geometry and in real life, by knowing the
  laws of logic

3
Conditional Statement

• A conditional statement has a hypothesis and a
  conclusion.
• If the weather is good, then I will go
  swimming.

hypothesis, p
conclusion, q
4
Negation

• The negation of a hypothesis or of a conclusion
  is formed by denying the original hypothesis or
  conclusion.

5
Contrapositive

• The contrapositive of a conditional statement is
  true if and only if the conditional statement is
  true.
• This law of logic is not true of the converse.
• The truth or falsity of the original statement
  has no bearing on the truth or falsity of the
  converse

6
Example 1 Writing a Contrapositive and Converse

• A. Identify the hypothesis and conclusion of the
  original statement.
• Original Statement
• If Polly says Hello, then Paul says Hello.
• B. Write the Contrapositive of the original
  statement
• C. Write the converse of the original statement

7
A.

• Identify the hypothesis and conclusion of the
  original statement.
• If Polly says Hello, then Paul says Hello.

hypothesis, p
conclusion, q
8
B.
If Polly says Hello, then Paul says Hello.
hypothesis, p
conclusion, q

• Write the Contrapositive of the original
  statement
• If Paul does not say Hello, then Polly does not
  say Hello.

Hypothesis, q
Conclusion, p
9
C.
If Polly says Hello, then Paul says Hello.
hypothesis, p
conclusion, q

• C. Write the converse of the original statement
• If Paul says Hello, then Polly says Hello.

Hypothesis, q
Conclusion, p
10

• Both the original statement and its
  contrapositive are false, the photo is a
  counterexample.
• You cannot tell whether the converse is true or
  false.
• If Paul says Hello, you dont know what, if
  anything, Polly will say.

11
Using Laws of Logic to Reason

• Two patterns of logical reasoning that use
  conditional statements are the Law of Syllogism
  and the Law of Detachment

• Law of Syllogism (also known as the Transitive
  Property of Conditionals)

• Law of Detachment (also known as Modus Ponens)

12
Example 2Using the Laws of Logic

• On August 15, Mike visited Norfolk, Virginia.
  Assuming that the following statements are true,
  can you conclude that Mike rode on the Drachen
  Fire?
• If Mike visits Norfolk, then he will go to Busch
  Gardens
• If Mike goes to Busch Gardens, then he will ride
  the Drachen Fire.

13
Solution to Example 2

• Let p, q, and r represent the following
• p Mike visits Norfolk
• q Mike goes to Busch Gardens
• r Mike rides the Drachen Fire

14
p Mike visits Norfolkq Mike goes to Busch
Gardens Example 2r Mike rides the Drachen
Fire

• If Mike visits Norfolk, then he will ride the
  Drachen Fire.
• Finally, b/c you are told that Mike visited
  Norfolk, you can conclude (by the Law of
  Detachment) that he rode the Drachen Fire.

15
Applying the Laws of Logic to Contrapositives

• Modus Tollens

16
Applying the Laws of Logic to ContrapositivesIf
Mike visits Norfolk, then he will ride the
Drachen Fire.

• Suppose you know that Mike didnt ride the
  Drachen Fire (r)
• If Mike did not ride the Drachen fire, then he
  did not go to Busch Gardens
• If Mike didnt go to Busch Gardens, then he
  didnt visit Norfolk.
• Using the laws of syllogism and detachment, you
  can conclude that Mike did not visit Norfolk (p)

17
Addressing Misconceptions

• The Law of Syllogism does not say that p (the
  original condition or fact) is known to be true!
  It simply states that if we agree that the
  statement is true and that the statement
  is true, then we must accept that the
  statement is true.

18
Example 3Using the Laws of Logic

• The angles in the chain of statements refer to
  the figure. Assume each statement is true.
  Explain how you can conclude the measure of angle
  6 115

19
Solution to Example 3
You are told that . This is p1
(the hypothesis of the first premise). B/C p1 is
true, it follows that p2 is true. B/C p2 is true
it follows that p3 is true. Using the same
reasoning, you can finally conclude that p6 is
true. Thus,

• By repeated use of the Laws of syllogism we have
• By the Law of Detachment, because

20
Solution to Example 3
given
supplementary angles
given
vertical angles
given
supplementary angles
21
The End