# Day 12 Deductive Reasoning

Shared by:
Categories
Tags
-
Stats
views:
4
posted:
9/29/2012
language:
English
pages:
17
Document Sample

```							Geometry Honors

Day 15
Deductive Reasoning
Today’s Objectives
 Daily Quiz
 Review Homework

 Deductive Reasoning

 Assignment
Standards:
   Make conjectures with justifications about
geometric ideas. Distinguish between information
that supports a conjecture and the proof of a
conjecture.
   Determine whether two propositions are logically
equivalent.
   Use methods of direct and indirect proof and
determine whether a short proof is logically valid.
   Write geometric proofs, including proofs by
geometry. Use and compare a variety of ways to
present deductive proofs, such as flow charts,
paragraphs, two-column, and indirect proofs.
Inductive vs. Deductive
Reasoning
 Recall that inductive reasoning bases its
conclusions on observations and patterns. It is
useful, but we can never prove a conjecture
definitively by just providing examples.
 Deductive reasoning uses facts, rules,
definitions, or properties to reach logical
conclusions from given statements.
Deductive Reasoning

   Deductive reasoning – a thought process in
which a conclusion is an inevitable logical
consequence of its premise.
 Example: 1) All men are mortal
2) Socrates is a man.
 What must be the conclusion?
3) Socrates is mortal.
   Deductive reasoning is powerful because, if
the starting point of your reasoning is true, the
conclusion has to be true.
Deductive Reasoning
   What is the premise, and what is the
conclusion of the following?
   If there is lightning, then the power will go off. If
the power goes off in the middle of the night, my
alarm won’t go off. If my alarm does not go off
in the morning, I won’t wake up on time. If I
don’t wake up on time, then I can’t make my
children breakfast. If my children don’t eat
breakfast, they will be cranky. If my oldest
daughter is cranky at school, she will hit
someone. If she hits someone, she will be
expelled. If she gets expelled, she won’t get
into college. If she doesn’t get into college, she
will be poor.
Deductive Reasoning
   Deductive reasoning is required to prove a
conjecture. There are various methods of valid
deductive reasoning.
   Note: Saying a conclusion is valid is different than
saying it is true. Valid means that the reasoning is
sound based on the given premise.
 The Law of Detachment states that if p  q
is a true statement, and p is true, then q must be
true.
 In other words, as long as the facts are true, then
the conclusion reached with deductive reasoning
will have to be true.
Examples
   Turn to p. 116 and study the examples.
Then determine whether the Guided
Practice problems show valid or invalid
reasoning.
The Law of Syllogism
 The Law of Syllogism states that if p  q and
q  r are true statements, then p  r is also a
true statement.
 When using the Law of Syllogism, the conclusion
of the first statement must be the hypothesis of the
next.
   You can stack as many of these together as you
wish. If they are properly set up, then the
hypothesis of the first statement will lead to the
conclusion of the last statement.
Examples
   Turn to p. 118, study the examples, and
complete the Guided Practice problems.
Postulates
 Geometry is an axiomatic system. In other words, it is
a system that is created by deductive reasoning,
starting from a few commonly accepted premises.
 Ideas that we accept without proof are called
postulates (or axioms).
 In a chain of deductive reasoning (cf. the Law of
Syllogism), each conclusion becomes the next premise.
But there must be an original premise that we start the
chain with. This original premise, by its nature, will
be unproven. This is where postulates come in.
Postulates (p. 125)
 We have discussed some postulates before:
 Through any two points, there is exactly one line.
 Through any three noncollinear points, there is exactly one
plane.
 A line contains at least two points.
 A plane contains at least three noncollinear points.
 If two points lie in a plane, then the entire line containing those
points lies in that plane.
 If two lines intersect, then their intersection is exactly one
point.
 If two planes intersect, then their intersection is a line.
 Note: The book numbers these postulates (2.1, 2.2, etc.), but
you don’t have to worry about the numbering.
Postulates and Reasoning
 Postulates can be used to support our reasoning process.
 Look at Example 2 on p. 126 and answer the Guided
Practice problems.
 When we solve problems, we should always be prepared
to justify our reasoning with the postulates or theorems
we are applying.
Introducing Proofs
 A proof is a detailed description of the logical process used
to deduce a fact (called a theorem) from previously known
information.
 A theorem is a conjecture that has been proven true by
deductive reasoning.
 Proofs allow us to make universal statements or rules that
apply to all situations of a given description.
 For example, how do we know that the Pythagorean Theorem
can be used on all right triangles?
 We don’t. Yet.
 A proof is a logical argument in which every statement is
supported by a reason that we have accepted as true.
 The reasons might be based off of definitions, mathematical
properties, common knowledge we accept without proof (i.e.,
postulates), or previously proven theorems.
Informal Proofs
 The proof process is as follows (p. 127):
 List the given information, and draw a diagram to illustrate the set-
up.
 State the theorem or conjecture to be proven.
 Create a deductive argument by forming a logical chain of statements
linking the given to what you are trying to prove.
 Justify each statement with a reason. (Can include given
information, definitions, algebraic properties, postulates, and
theorems.)
 State what you have proven.
 An informal proof (or paragraph proof) involves writing in
paragraph form an explanation as to why a conjecture is true.
 Let’s go over Example 3 and Guided Practice 3 on p. 127.
 Now that we’ve proven this idea, we can call it a theorem. (Your book
calls it the Midpoint Theorem, but I am less concerned with names
than with understanding the concept behind them.)
Can you…?
 Distinguish between inductive and deductive
reasoning?
 Use the Laws of Detachment and Syllogism to
determine if deductive reasoning is valid?
 Support deductive reasoning using postulates,
theorems, definitions, arithmetic properties, etc?
 Follow and understand a simple informal proof?
Homework 7

 Workbook, pp. 21, 24
 Book, p.122, #35-40, 42, 43, 45

```
Related docs
Other docs by HC120929092644
STATE FFA NURSERY/LANDSCAPE CONTEST
Course Support