Alice in Wonderland does Geometry?
Come, we shall have some fun now! thought Alice. I’m glad they’ve begun asking riddles. I believe I
can guess that, she added aloud.
Do you mean that you think you can find out the answer to it? said the March Hare?
Exactly so, said Alice.
Then you should say what you mean, the March Hare went on.
I do, Alice hastily replied; at least . . . at least I mean what I say . . . that’s the same thing, you know.
Not the same thing a bit! said the Hatter. You might just as well say that “I see what I eat” is the
same thing as “I eat what I see.”
You might just as well say, added the March Hare, that “I like what I get” is the same thing as “I get
what I like.”
You might just as well say, added the Dormouse, who seemed to be talking in his sleep, that “I
breathe when I sleep” is the same thing as “I sleep when I breathe!”
It IS the same thing with you, said the Hatter, and here the conversation dropped, and the party sat
silent for a minute, while Alice thought over all she could remember about ravens and writing-desks,
which wasn’t much.
What kind of logic statements are included above?
What Can You Say?
Conditional:
If I live in England, then I live in Europe.
Converse:
If I live in Europe, then I live in England.
Inverse:
If I do not live in England, then I do not live in
Europe.
Contrapositive: If I do not live in Europe, then I do not live in
England.
What do all four statements have in common?
2-1 Conditionals Statements
An IF, THEN statement is called a
conditional statement or an implication.
The IF part is called the hypothesis; the THEN part is
the conclusion.
2-1 Conditionals Statements
Conditional
If a candy bar is a Milky Way, then it contains caramel.
hypothesis
conclusion
A conditional statement can be TRUE or FALSE (truth value).
Use a counterexample to show a conditional is false.
• If it is January, then the month has 31 days.True
• If it is February, then the month has 28 days.False
counterexample: leap year
Converse
Switch the hypothesis and the conclusion.
Some examples:
Conditional:
Converse:
If an animal is a Maine Coon, then it is a cat.
If an animal is a cat, then it is a Maine Coon.
Conditional:
Converse:
If a figure is a rhombus, then it has four sides. True
If a figure has four sides, then it is a rhombus. False
True
False
Special Notation
Conditional:
If p, then q is represented as p  q
(or p implies q).
Converse:
If q, then p is represented as q  p
(or q implies p).
What Can You Say?
True
False
Conditional:
Converse:
If an animal is a hen, then it lays eggs.
If an animal lays eggs, then it is a hen.
Conditional:
If an angle is acute, then its measure is
less than 90.
If an angle has a measure less than 90,
then it is acute.
If x=–10, then x2 = 100 True
If x2 = 100, then x=–10. False
Converse:
Conditional:
Converse:
SOME ARE TRUE, WHILE OTHERS ARE NOT!!!!
True
True
Change these statements into conditionals
which are converses of each other.
Alice: I say what I mean = I mean what I say?
If I say it, then I mean it.
If I mean it, then I say it.
Mad Hatter: I see what I eat = I eat what I see?
If I see it, then I eat it.
If I eat it, then I see it.
March Hare: I like what I get = I get what I like?
If I like it, then I get it.
If I get it, then I like it.
Dormouse: I breathe when I sleep = I sleep when I breathe?
If I am breathing, then I am sleeping.
If I am sleeping, then I am breathing.
Venn Diagram
You can use a Venn Diagram to determine if the
associated statements are true or false. Draw
a Venn Diagram for the following conditional.
“If Mark is a member of the CRHS soccer
team, then he is a member of the CRHS
student body.”
Converse is not true.
Soccer
Team
Mark
CRHS Student
Body
“If I live in Germany, then I live in Europe.”
“If I live in Europe, then I live in Germany.”
Germany
Europe
2-2 Biconditionals and Definitions
Biconditional: a statement resulting when a conditional and its converse are
BOTH true.
If p  q true AND q  p true, then p  q
Examples:
Conditional:
If a figure is a triangle, then it has exactly 3 sides.
Converse:
If a figure has exactly 3 sides, then it is a triangle.
Biconditional:
A figure is a triangle if and only if it has exactly three
sides
OR
A figure has exactly 3 sides if and only if it is a triangle.
cont.
Definition:
A description of an item that is precise; it is never
vague or ambiguous. Definitions are always reversible.
Example:
Reversible
Definition:
A right angle is an angle whose measure is 90.
Conditional:
If an angle is a right angle, then its measure is 90
Converse:
If an angle measures 90, then it is a right angle.
Biconditional:
An angle is a right angle if and only if its measure is 90.
True
True
OR
An angle measures 90 if and only if it is a right angle.
Geometry is based on a DEDUCTIVE STRUCTURE 
conclusions are justified by assumed or proved
statements.
Undefined Terms
point
line
Postulates (***NOT always reversible***)
unproved assumptions
Definitions (***definitions are reversible***)
acute angle
right angle, etc.
Theorems (***NOT always reversible***)
mathematical statements that can be proved
2-3 Deductive Reasoning
Law of Detachment: If a conditional is true and its
hypothesis is true, then its conclusion is true.
If p  q is true AND p is true, then q is true.
Example 1
A gardener know that if it rains, the garden will be
watered. It is raining. The garden will be watered.
Example 2
If you make a field goal in basketball, you score two points.
Jenna scored two points in basketball. Jenna made a field
goal. Not possible. Does NOT use the Law of Detachment.
2-3 cont.
Law of Syllogism (Chain of Reasoning):
If p  q is true and if q  r is true, then p  r is true.
Like the transitive property!
Example 1
If I practice soccer, I will score more goals for JV. If I
score more goals for JV, I will move up to Varsity.
Conclusion: If I practice soccer, I will move up to Varsity.
2-3 cont.
Example 2
If a quadrilateral is a square, then it contains four right
angles. If a quadrilateral contains four right angles, then
it is a rectangle.
Conclusion: If a quadrilateral is a square, then it is a
rectangle.
Example 3
If it rains, then Jan stays inside. If Jan stays inside, then
she does not get wet.
Conclusion: If it rains, then Jan does not get wet.
2-3 cont.
Combining Detachment and Syllogism
If the circus is in town, then there are tents at the
fairground. If there are tents at the fairground, then Paul
is working as a night watchman.
The circus is in town.
Paul is working as a night watchman.