Day 12 Deductive Reasoning

							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
    contradiction and proofs involving coordinate
    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