Download Geometry Section 1.4 Notes

1.4
Introduction to
Deductive Proofs
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Slide 1
Definition
A conditional statement is a statement that is
written, or that can be written, in “if-then” form.
When a statement is in “if-then” form, the phrase
that follows “if” is called the hypothesis, and the
phrase that follows “then” is called the conclusion.
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Slide 2
Definition
If a figure is a pentagon, then it has five sides.
Hypothesis
Conclusion
If it is snowing, then it is cloudy.
Hypothesis
Conclusion
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Slide 3
Symbols
When working with conditional statements, it is
often handy to use shortcut notations.
You may see any of the following:
Hypothesis
Conclusion
if
p
then
q
p
implies
q
p
q

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Slide 4
Identifying the Hypothesis and the
Example
Conclusion
Identify the hypothesis (p) and the conclusion (q).
a. If an animal is a turtle, then the animal is a reptile.
Solution
Hypothesis (p): an animal is a turtle
Conclusion (q): the animal is a reptile
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Slide 5
Identifying the Hypothesis and the
Example
Conclusion
Identify the hypothesis (p) and the conclusion (q).
b. If a number is even, then the number is not odd.
Solution
Hypothesis (p): a number is even
Conclusion (q): the number is not odd
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Slide 6
Example
Writing a Conditional Statement
A hypothesis (p) and a conclusion (q) are given. Use
them to write a conditional statement, p  q.
a. p: a figure is a square
q: the figure is not a triangle
Solution
If a figure is a square, then the figure is not a
triangle.
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Slide 7
Example
Writing a Conditional Statement
A hypothesis (p) and a conclusion (q) are given. Use
them to write a conditional statement, p  q.
b. p: 9 is a perfect square
q: 9 is not a prime number
Solution
If 9 is a perfect square, then 9 is not a prime
number.
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Slide 8
Example
Writing a Conditional Statement
Write the following statement in “if-then” form.
Acute angles measure less than 90°.
Solution
If an angle is acute, then it measures less than 90°.
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Slide 9
Conditional Statements
A conditional statement may be either true or false.
• A conditional statement is true if every time the
hypothesis is true, then the conclusion is also true.
• A conditional statement is false if there is a
counterexample in which the hypothesis is true, but
the conclusion is false.
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Slide 10
Is a Conditional Statement True
Example
or False?
Determine whether each conditional statement is
true or false.
a. If an angle measures 92°, then it is
an obtuse angle.
Solution
This conditional statement is true. All angles that
measure 92° are obtuse angles.
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Slide 11
Is a Conditional Statement True
Example
or False?
Determine whether each conditional statement is
true or false.
b. If a month begins with the letter J, then the month
has 31 days.
Solution
This conditional statement is false. The month June
starts with a J but has 30 days.
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Slide 12
Negation
The negation of a statement is formed by writing the
negative of the statement. (Note: The notation for negation
is =, so ~p is read “not p.”)
Statement
Negation
The computer cover is red. The computer cover is not red.
The rarest blood group for
humans is group AB.
The rarest blood group for
humans is not group AB.
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Slide 13
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Slide 14
Equivalent Statements
If two statements are both always true or both
always false, we call them equivalent statements.
• The conditional statement and the contrapositive
statement in the table are both true and are examples
of equivalent statements.
• The converse statement and the inverse statement
are both false and are examples of equivalent
statements.
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Slide 15
Writing Related Conditional
Example
Statements
Write the (a) converse, (b) inverse, and (c)
contrapositive of the given conditional statement.
If it is raining, then it is cloudy.
p
q
Solution
Converse: If it is cloudy, then it is raining.
q
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p
Slide 16
Writing Related Conditional
Example
Statements
Write the (a) converse, (b) inverse, and (c)
contrapositive of the given conditional statement.
If it is raining, then it is cloudy.
p
q
Solution
Inverse: If it is not raining, then it is not cloudy.
p
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q
Slide 17
Writing Related Conditional
Example
Statements
Write the (a) converse, (b) inverse, and (c)
contrapositive of the given conditional statement.
If it is raining, then it is cloudy.
p
q
Solution
Contrapositive:
If it is not cloudy, then it is not raining.
p
q
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Slide 18
Deductive Reasoning
Deductive reasoning is the process of proving a
specific conclusion from one or more general
statements. A conclusion that is proved true by
deductive reasoning is called a theorem.
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Slide 19
Example
Giving Reasons for Statements
Solve 5x + 12 = 47. Give a reason to justify each
statement.
Solution
Statements
5 x  12  47
5 x  35
x7
Reasons
Given
Subtraction Property of Equality
Division Property of Equality
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Slide 20
Example
Giving Reasons for Statements
Solve 7x – 10(5 + 3x) = 2x. Give a reason to justify
each statement.
Solution
Statements
7x – 10(5 + 3x) = 2x
Reasons
Given
7 x  50  30 x  2 x
23x  50  20
50  25x
2  x
Multiply or Distributive Property
Simplify or Combine like terms
Addition Property of Equality
Division Property of Equality
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Slide 21
Proof Basics
A proof is an argument that uses logic to establish
the truth of a statement. There are many formats for
proofs, but for now, we will use a two-column
proof.
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Slide 22
Example Writing a Direct Proof
Write a direct proof.
Given: Given
Prove: Conclusion
Proof:
Statements
Reasons
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Slide 23
Example Writing a Direct Proof
Page 32 Exercise 2
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Slide 24