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Geometry
Stuff that should be in 2.2 and
2.3
1. Counterexample – an example used to
show something is false.
2. A syllogism is also known as
DEDUCTIVE REASONING.
Inverse Statement
Negate the hypothesis and conclusion of the
conditional statement.
Conditional:
If it’s hot outside, then the sun is shining.
Inverse:
If it’s not hot outside, then the sun is not shining.
Contrapositive Statement
SWITCH the hypothesis and conclusion of the
INVERSE statement.
Conditional:
If it’s hot outside, then the sun is shining.
Inverse:
If it’s not hot outside, then the sun is not shining.
Contrapositive:
If the sun is not shining, then it is not hot outside.
Notation
1.
2.
3.
4.
Conditional statement
ab
Converse statement
ba
Inverse statement
~a  ~b
Contrapositive statement ~b  ~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 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 angle. 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
Which ones are
Floppers?
Floppers
Not 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.
♥ 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
B
C
Assignment
• pg 95, write inverse and contrapositive for
#9 and #17
• pg 103, 8 & 9, 17-27, 30-38