Ch 2: Reasoning and Proof

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Ch 2 Reasoning and Proof

• A proof is a convincing logical argument that
  uses deductive reasoning.

• Types of statements used in a logical argument
• Conditional statements
• Bi conditional statements
• Definitions
• Properties
• Postulates
• Theorems

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Deductive Reasoning
GEOMETRY LESSON 2-3
(For help, go to the Lesson 2-1.)
Write the converse of each statement. 1. If you
dont sleep enough, then your grades
suffer. 2. If you want to arrive on time, then
you must start early. Write each statement as a
conditional. 3. Leap years have 366
days. 4. Students who do not complete their
homework will have lower grades. 5. Two lines
that are perpendicular meet to form right
angles. 6. Every sixteen year old is a teenager.
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Deductive Reasoning
GEOMETRY LESSON 2-3
1. Switch the hypothesis and conclusion If your
grades suffer, then you dont sleep
enough. 2. Switch the hypothesis and conclusion
If you must start early, then you want to arrive
on time. 3. Rewrite the statement as an if-then
statement If a year is a leap year, then it has
366 days. 4. Rewrite the statement as an if-then
statement If students do not complete their
homework, then they will have lower
grades. 5. Rewrite the statement as an if-then
statement If two lines are perpendicular, then
they meet to form right angles. 6. Rewrite the
statement as an if-then statement If a person is
16 years old, then that person is a teenager.
Solutions
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Deductive Reasoning

• Is also called logical reasoning.
• It is the process of developing a proof of a
  statement by building upon already proven
  statements.
• If the statements in the proof are true, then the
  conclusion will also be true.

Postulates, Theorems, definitions
Begins with given info
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Deductive Reasoning
GEOMETRY LESSON 2-3
A gardener knows that if it rains, the garden
will be watered. It is raining. What conclusion
can he make?
It is raining
Given
The garden will be watered.
If it rains, the garden will be watered.
QED
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Deductive Reasoning
GEOMETRY LESSON 2-3
For the given statements, what can you conclude?
Definition If ?A is acute, m?A ?A is acute
?A is acute
Given
Definition of acute angle.
m? AQED
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The Law of Detachment
If a conditional is true and its hypothesis is
true, then the conclusion is also true.
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Does the following argument illustrate the Law of
Detachment?
If a conditional is true and its hypothesis is
true, then the conclusion is also true
Given If you make a dunk shot in basketball,
you score two points.Jenna scored two points in
basketball.
You conclude Jenna made a dunk shot. The two
given statements mean that a conditional and its
conclusion are both true. The Law of Detachment
applies only if a conditional and its hypothesis
are true. You can make no conclusion. You cannot
conclude that Jenna made a dunk shot.
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The Law of Syllogism
If p? q and q ?r, then p ? r. If the conclusion
of the first conditional is the hypothesis of the
second conditional, then the first hypothesis
implies that the second conclusion will be true.
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Use the Law of Syllogism to draw a conclusion
from the following true statements
If p? q and q ?r, then p ? r.
If a quadrilateral is a square, then it contains
four right angles. If a quadrilateral contains
four right angles, then it is a rectangle.
The conclusion of the first conditional is the
hypothesis of the second conditional. This means
that you can apply the Law of Syllogism. So you
can conclude If a quadrilateral is a square,
then it is a rectangle.
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Use the Laws of Detachment and Syllogism to draw
a possible conclusion.
If the circus is in town, then there are tents at
the fairground. If there are tents at the
fairground, then Paul is working as a night
watchman. The circus is in town.
The circus is in town.
Given
There are tents at the fairground.
Law of Detachment.
Law of Syllogism.
Paul is working as night watchman.
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Re Cap
A proof is a convincing logical argument that
uses deductive reasoning. Law of Detachment If a
conditional is true and its hypothesis is true,
then the conclusion is also true. Law of
Syllogism If p? q and q ?r, then p ? r.
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Additional Practice
GEOMETRY LESSON 2-3
Use the three statements below. A. If games are
canceled, then Maria reads a book. B. If it
snows, then games are canceled. C. It is
snowing. 1. Using only statements A and B, what
can you conclude? 2. Using only statements B
and C, what can you conclude? 3. Using
statements A, B, and C, what can you
conclude? 4. Suppose both statement B and
games are canceled are true. Can you conclude
that statement C is true? Explain.
If it snows, then Maria reads a book.
Games are canceled.
Maria is reading a book.
No sample you cannot apply the Law of
Detachment.
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