Honors
Geometry
Chapter 2, Section 1
Inductive Reasoning and Conjecturing
What do you think?
A B C D E F G ___
What would you expect to see next?
This educated guess is called a conjecture!
A B C D E F G
H
In this case, we have a counter-example, that shows our
original conjecture to be false.
You try it!!
Given: Points A, B, & C with AB = 15,
BC = 12 & AC = 9
Make a conjecture.
Some possible conjectures:
A, B and C are non-collinear.
A, B and C form the vertices of a triangle.
One More Time!!
Sam was driving home from
school and her car died. Make a
conjecture about why her car
died and then determine a way
for Sam to test each conjecture.
And yet another! !
Given that points P, Q & R are collinear, Ed
made the conjecture that Q is between
P & R. Determine if this conjecture is true
or false. Explain your answer.
Honors
Geometry
Chapter 2, Section 2
If, then statements and Postulates
If, Then Statements
If we score more points, then we will win.
hypothesis
conclusion
You
try
it
!
!
Identify the hypothesis and conclusion in the following
statements:
chickens
have
lips
pigs
fly
1) If chickens have lips, then pigs fly.
∠
2 is a right ∠
m∠2 = 90°
2) If m∠2 = 90°, then ∠2 is a right ∠.
I will drive
I get my license
3) I will drive, if I get my license.
4) A (Hint:
cautious
person
wears
their
Try writing
in If ... , then
... form
first!) seat belt.
If a person is cautious, then they wear their seat belt.
Conditional
Statements
If–then statements are also known as
conditional statements. This is because we
claim that the truth (or falsehood) of the
conclusion is conditional upon the truth (or
falsehood) of the hypothesis.
Some Terminology
If A, then B can also be written as A ⇒ B
~A is the opposite of A
Example: A is “Dogs are animals.”
So: ~A is “Dogs are not animals.”
Converse
Converse of a Conditional Statement
Statement: A ⇒ B
Converse: B ⇒ A
Statement: If fish swim, then cotton is soft.
Converse: If cotton is soft, then fish swim.
A⇒B
B⇒A
Practice !
State the converse of the statement below:
If I drive too fast, then I will get a ticket.
Converse: If I got a ticket, then I was driving too fast.
Inverse
Inverse of a Conditional Statement
Statement: A ⇒ B
Inverse: ~A ⇒ ~B
Statement: If fish swim, then cotton is soft.
A⇒B
Inverse: If fish do not swim, then cotton is not soft. ~A ⇒ ~B
Practice !
State the inverse of the statement below:
If I drive too fast, then I will get a ticket.
Inverse:
If I do not drive too fast, then I will not get a ticket.
Contrapositive
Contrapositive of a Conditional Statement
Statement: A ⇒ B
Contrapositive: ~B ⇒ ~A
Statement: If fish swim, then cotton is soft.
A⇒B
Converse: If cotton is not soft, then fish do not swim.
~B ⇒ ~A
Practice !
State the Contrapositive of the statement below:
If I drive too fast, then I will get a ticket.
Contrapositive:
If I did not get a ticket, then I was not driving too fast.
More Postulates
Postulate 2–1
Postulate 2–2
Postulate 2–3
Postulate 2–4
Postulate 2–5
Postulate 2–6
Through any two points there is exactly one line.
Through any three non-collinear points there is exactly one plane.
A line contains at least two points.
A plane contains at least three non-collinear points.
If 2 pts. lie in a plane, then the entire line containing those pts. . . .
lies in the plane.
If two planes intersect, then their intersection is a line.
Try it !!!
The two angles that form a linear pair are
supplementary.
Write in if–then form
If two angles form a linear pair, then they are supplementary.
then write the converse
If two angles are supplementary, then they form a linear pair.
Bi-Conditional
Statements
A true if - then statement whose converse is also true
can be written as a bi-conditional statement.
Example
If a man is unmarried, then he is a bachelor.
If a man is a bachelor, then he is unmarried.
becomes
A man is unmarried, if and only if, he is a bachelor.
or
A man is unmarried, iff, he is a bachelor.
Bi-Conditional Statements
2
A right angle has a measure of 90°.
Write as an if, then statement.
If an angle is a right angle, then its measure is 90°
Write
its
converse.
If an angle has a measure of 90°, then it is a right
angle
Write as a bi-conditional statement.
An angle is a right angle, iff, its measure is 90°
Assignment
2.2 Practice Worksheet
Download

Honors Geometry