The student will be able to: by IlsqWsUD

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									                 Geometry
             Chapter 2 section 3
              Pages #98 - 100
• Biconditionals and Definitions.

• Standard addressed:
  Construct and judge the validity of a logical argument and
  give counterexamples to disprove a statement.




                    Lesson 2-3: Biconditionals and Definitions   1
     The student will be able to:
• Write Biconditional Statements
• Recognize good definitions




               Lesson 2-3: Biconditionals and
                                                2
                        Definitions
                    Review Notation
     Let a = the figure is a triangle
                                     Let b = it has three sides

1.    Conditional statement     a  b (If a, then b)
2.    Converse statement        b  a (If b, then a)
3.    Inverse statement        ~a  ~b (If not a, then not b)
4.    Contrapositive statement ~b  ~a (If not b, then not a)
               Example/Practice
• Using the sentence:           “My dog has fleas”,
   write the (a) conditional, (b) converse, (c)
   inverse and (d) contrapositive statements.
a) If it is my dog, then it has fleas
b) If it has fleas, then it is my dog.
c) If it is not my dog, then it does not have fleas.
d) If it does not have fleas, then it is not my dog.
                      Biconditional
  • When a conditional statement and its converse are both true,
    the two statements may be combined into a true biconditional
    statement.
  • Use the phrase if and only if

Statement: If an angle is right angle, then it has a measure of 90.

Converse: If an angle has a measure of 90, then it is a right angle.

Biconditional: An angle is right angle if and only if it has a measure of
               90.


                           Lesson 2-3: Biconditionals and
                                                                        5
                                     Definitions
            Biconditional Statement
When a conditional statement AND the converse are BOTH TRUE, this
     creates a special case called ‘biconditional”.
Conditional:
If a quadrilateral has 4 right angles, then it is a rectangle.       ab
                                                                 (true)
Converse:
If it is a rectangle, then it is a quadrilateral with 4 right angles. ba
                                                                 (true)
Biconditional:
A quadrilateral has 4 right angles if and only if it is a rectangle.
     (don’t use if and then)                                      a b
                                                          (true BOTH ways)
                          iff means “if and only if”
A biconditional is a statement that is true backwards and forwards.
A biconditional is a DEFINITION.
                              Floppers                      a         b



                                                            c


                                                             d        e




Floppers                                                  Which ones are
                                                          Floppers?
                             Not Floppers              Only b and c are floppers
• Let’s write a definition:
Step 1: write a conditional statement:
If a figure is a Flopper, then it has one eye and two tails. (true)
Step 2: write the converse:
If a figure has one eye and two tails, then it is a Flopper. (true)
Step 3: write the biconditional (definition)
A figure is a Flopper if  and only if it has one eye and two tails.
Or   A figure is a Flopper iff it has one eye and two tails.
Quick Review

If the original conditional is true AND the
   converse is true, then the statement is a
   definition.
This statement is called a BICONDITIONAL
Notation: p q (note the double arrow)
We say: “p if and only if q”
This can be abbreviated to: p iff q
               Adjacent Angles
          What does ‘adjacent’ mean?
    • Adjacent angles are angles that share a vertex
      and a side. They do not share any interior
      points. (They don’t overlap.)
A           B
                            AXB & BXC are adjacent
                 C          angles. They share the vertex X
                            and the ray XB.

      X                D
                            AXC & BXC are NOT adjacent
                            angles. They share the vertex X
                            but they overlap thus causing the
                            sharing of interior points.
                  Practice
• Yes or no: are these adjacent angles?
A
                    No these angles
                    do not share a vertex
                    or a side




B    Yes                               C
                                       NO these angles
                                       share a side but
                                       not a vertex
         Biconditional - Example




Symbology:   pq       “p if and only if q”


                   Lesson 2-3: Biconditionals and
                                                    11
                             Definitions
             Biconditional - Example

Biconditional: A ray is an angle bisector if and only if it
divides an angle into two congruent angles.
 p: A ray is an angle bisector.
 q: A ray divides an angle into two congruent angles.
p  q : If a ray is an angle bisector, then it divides an angle into two congruent angles.

q  p : If a ray divides an angle into two congruent angles then, it is an angle bisector, .




                               Lesson 2-3: Biconditionals and
                                                                                       12
                                         Definitions
                Definitions
A good definition has the following components:
  - Uses clearly understood terms.
  - Is precise (avoids words such as large, sort
    of, some, etc.).
  - Is reversible. You can write it as a true
    biconditional.


                 Lesson 2-3: Biconditionals and
                                                  13
                           Definitions
Definitions




Lesson 2-3: Biconditionals and
                                 14
          Definitions
Definitions




Lesson 2-3: Biconditionals and
                                 15
          Definitions
           Objective Practices
• Lesson Check: pg. 101 #1-6
  – Check if off with me when finished before the
    end of the period


• HW pgs. 101 – 104 #7 – 45 odds


     DUE Thursday Nov 10

								
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