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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
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The student will be able to:
• Write Biconditional Statements
• Recognize good definitions
Lesson 2-3: Biconditionals and
Definitions
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Review Notation
Let a = the figure is a triangle
Let b = it has three sides
1.
2.
3.
4.
Conditional statement
a  b (If a, then b)
Converse statement
b  a (If b, then a)
Inverse statement
~a  ~b (If not a, then not b)
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
Definitions
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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
Floppers
Not Floppers
e
Which ones are
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
Or
and only if it has one eye and two tails.
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
angles. They share the vertex X
and the ray XB.
C
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
Definitions
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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
Definitions
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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
Definitions
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Definitions
Lesson 2-3: Biconditionals and
Definitions
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Definitions
Lesson 2-3: Biconditionals and
Definitions
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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