Download Learner Objective: Students will write the converse, inverse, and

Advanced Geometry
Section 1.7 and 1.8- Deductive Structure / Statements of Logic
Learner Objective: Students will write the converse, inverse, and contrapositive of a
conditional statement, determine the truth value of statements and will use the chain rule to
make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
1
A
B
C
D
E
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Building blocks of Geometry:
Undefined Terms - Terms which we can't officially define. We describe their
properties but don't formally define them.
examples: point, line, plane
Definitions - State the meaning of a term. There is never a need to prove a
definition. (It is true because we say it is true.)
examples: Def. of a right angle, Def. of a segment bisector...
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Building blocks of Geometry (cont.):
Postulates - An unproved assumption. These are pretty obvious statements which we
are unable to prove but which clearly must be true.
example: Two points determine a line.
Theorems - A mathematical statement which can be proved.
example: If two angles are both right angles, then they are congruent.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Definitions, Postulates and Theorems are all conditional statements. They have some
(sufficient) condition that leads to a (necessary) conclusion.
If two angles are both right angles,
Having two right angles is sufficient evidence to conclude that the angles are
congruent.
then they are congruent.
When two angles are both right angles, it is necessary to conclude that they are
congruent (they have to be).
What happens if we reverse the condition and conclusion?
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Here is the theorem with the condition and conclusion reversed:
If two angles are congruent
Is knowing that two angles are congruent sufficient evidence to conclude that
they must be right angles?
then they are both right angles.
If two angles are congruent, is it necessary to conclude that they are right angles?
Is this a true statement?
For a statement to be considered true, it must ALWAYS be true. NOT ALWAYS TRUE =
FALSE
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Conditional statements can be written in "If p then q" form where p is the condition or
hypothesis of the statement (trigger) and q is the conclusion (result).
"If p then q" can be symbolized
(also read "p implies q")
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Reversing the condition and the conclusion results in the Converse of the original
statement. Thus the convers is symbolized
If both the conditional and the converse are true, the statement is said to be
reversible.
Definitions are always reversible.
Postulates and Theorems are sometimes reversible.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
The Negation (opposite) of the statement p is "not p" and is symbolized
The Inverse of the conditional statement is formed when we negate both p and q.
Thus the inverse is symbolized
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
The Contrapositive of the conditional statement is formed by both reversing and
negating p and q. Thus, the contrapositive can be symbolized
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
The conditional statement and it's contrapositive have the same "truth value". That is,
they are either both true or both false.
The inverse and the converse have the same truth value.
Conditional
T
T
F
F
Converse
T
F
T
F
Inverse
T
F
T
F
Contrapositive
T
T
F
F
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Example
Conditional:
Converse:
Inverse:
Contrapositive:
If two angles are both right angles, then they are
congruent
If two angles _______________
then ______________________
If two angles _______________
then ______________________
If two angles _______________
then ______________________
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Example
Conditional:
If an angle is an acute angle,
then its measure is greater than 0 and less than 90.
Converse:
If an angle _____________________________,
then __________________________________.
Inverse:
If an angle _____________________________,
then __________________________________.
Contrapositive:
If an angle _____________________________,
then __________________________________.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
A series of conditional statements can be connected together using the Chain Rule.
If
and
then
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
Example:
,
,
, and
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
At Hilldale High School, there is a rule that any student caught fighting must be given a three day
suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are true?
Bill has been given a three day suspension so we know that he must have been caught
fighting.
Bob has never been caught fighting, so we know that he has
day suspension.
Bo has never been given a three day suspension, so we know
been caught fighting.
never been given a three
that he has never
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
At Hilldale High School, there is a rule that any student caught fighting must be given a three day
suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are
true?
Conditional:
If fighting then suspension.
Bill has been given a three day suspension so we know that he must have been caught
fighting.
Converse:
False
If suspension then fighting
It is not necessary for someone to Fight in order to receive a suspension. Bill could
have been suspended for something else. The converse is false.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
At Hilldale High School, there is a rule that any student caught fighting must be given a three day
suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are
true?
Conditional:
If fighting then suspension.
Bob has never been caught fighting, so we know that he has
day suspension.
Inverse:
False
never been given a three
If NOT fighting then NO suspension.
Fighting is a sufficient condition to receive a suspension but not a necessary one. Bob
could have been suspended for something else. The inverse is false.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
At Hilldale High School, there is a rule that any student caught fighting must be given a three day
suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are
true?
Conditional:
If fighting then suspension.
Bo has never been given a three day suspension, so we know
been caught fighting.
Contrapositive:
True
that he has never
If NO suspension then NOT fighting.
Since suspension is a necessary result of fighting, NOT being suspended is sufficient
to conclude that there was NO fighting. The contrapositive is true.
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth
Learner Objective: Students will write the converse, inverse, and contrapositive of a
value of statements and will use the chain rule to make logical conclusions.
conditional statement, determine the truth