Download GEOMETRY 2.1 Conditional Statements

GEOMETRY
2.1 Conditional Statements
and distance.
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2.1 CONDITIONAL STATEMENTS
ESSENTIAL QUESTION
When is a conditional statement true
or false?
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2.1 CONDITIONAL STATEMENTS
WHAT YOU WILL LEARN
oWrite conditional statements.
oUse definitions written as conditional statements.
oWrite biconditional statements.
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2.1 CONDITIONAL STATEMENTS
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2.1 CONDITIONAL STATEMENTS
CONDITIONAL
A type of logical statement that
has two parts, a hypothesis and
a conclusion.
A conditional can be written in
IF-THEN form.
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2.1 CONDITIONAL STATEMENTS
SHORTHAND
If HYPOTHESIS, then CONCLUSION.
If P, then Q.
In the study of logic, P’s and Q’s are universally
accepted to represent hypothesis and conclusion.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 1
If I study hard,
hard then I will get good grades.
HYPOTHESIS
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CONCLUSION
2.1 CONDITIONAL STATEMENTS
CAN YOU IDENTIFY THE HYPOTHESIS AND
CONCLUSION?
If today is Monday, then tomorrow is Tuesday.
Hypothesis: today is Monday
Conclusion: tomorrow is Tuesday.
Note: IF is NOT part of the hypothesis, and THEN is
NOT part of the conclusion.
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2.1 CONDITIONAL STATEMENTS
YOUR TURN
Underline the hypothesis and circle the conclusion.
1. If the weather is warm, then we should go swimming.
2. If you want good service, then take your car to Joe’s Service Center.
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2.1 CONDITIONAL STATEMENTS
REWRITING STATEMENTS.
oUse common sense.
The hypothesis always follows “IF.”
oDon’t over analyze it.
No “if?” The first part is usually the hypothesis.
oMake sure the sentence is grammatically correct.
Make your English teacher proud!
Does it sound right?
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 2A
Rewrite the following statement in if-then form:
All birds have feathers.
What is the hypothesis?
All birds
What is the conclusion?
have feathers
If-then form?
If an animal is a bird, then it has feathers.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 2B
Rewrite the following statement in if-then form:
You are in Texas if you are in Houston.
What is the hypothesis?
You are in Houston
What is the conclusion?
You are in Texas
If-then form?
If you are in Houston, then you are in
Texas.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 2C
Rewrite the following statement in if-then form:
An even number is divisible by 2.
What is the hypothesis?
An even number
What is the conclusion?
Divisible by 2.
If-then form?
If a number is even, then it is divisible by 2.
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2.1 CONDITIONAL STATEMENTS
YOUR TURN
Rewrite the conditional statement in if-then form.
3. Today is Monday if yesterday was Sunday.
If yesterday was Sunday, then today is
Monday.
4. An object that measures 12 inches is one foot long.
If an object measures 12 inches, then it is one
foot long.
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2.1 CONDITIONAL STATEMENTS
NEGATION
The negative of the original statement.
Examples:
I am happy.
I am not happy.
mC = 30°.
mC  30°.
A, B and C are on the same line.
A, B and C are not on the same line.
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2.1 CONDITIONAL STATEMENTS
NEGATION
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 3
Write the negation of each statement.
a. The ball is red.
The ball is not red.
b. The cat is not black.
The cat is black.
c. The car is white.
The car is not white.
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2.1 CONDITIONAL STATEMENTS
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2.1 CONDITIONAL STATEMENTS
RELATED CONDITIONAL STATEMENTS
Looking at the conditional statement:
If p, then q.
There are three similar statements we can make.
o Converse
o Inverse
o Contrapositive
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2.1 CONDITIONAL STATEMENTS
CONVERSE
If Q, then P.
The converse of a statement is formed by
switching the hypothesis and the conclusion.
Conditional:
If you play drums, then you are in the band.
Converse:
If you are in the band, then you play drums.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 4
Write the converse of the statement below.
If you like tennis, then you play on the tennis team.
Answer:
If you play on the tennis team, then you like tennis.
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2.1 CONDITIONAL STATEMENTS
INVERSE
If not P, then not Q.
The inverse is formed by taking the negation
of the hypothesis and of the conclusion.
Conditional:
If x = 3, then 2x = 6.
Inverse:
If x  3, then 2x  6.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 5
Write the inverse of the statement below.
If today is Monday, then tomorrow is Tuesday.
Answer:
If today is not Monday, then tomorrow is not
Tuesday.
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2.1 CONDITIONAL STATEMENTS
CONTRAPOSITIVE
If not Q, then not P.
The contrapositive is formed by switching and negating
the hypothesis and the conclusion.
(Take the inverse of the converse, or, the converse of the
inverse.)
Conditional:
If I am in 10th grade, then I am a sophomore.
Contrapositive:
If I am not a sophomore, then I am not in 10th grade.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 6
Write the contrapositive of the statement below.
If x is odd, then x + 1 is even.
Negate
Negate
x + 1 is not even
x is not odd
If x+1 is not even, then x is not odd.
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2.1 CONDITIONAL STATEMENTS
LOGICAL STATEMENTS
If I live in Mesa, then I live in Arizona.
Converse: (switch hypothesis and conclusion)
If I live in Arizona, then I live in Mesa.
Inverse: (negate hypothesis and conclusion)
If I don’t live in Mesa, then I don’t live in Arizona.
Contrapositive: (switch and negate both)
If I don’t live in Arizona, then I don’t live in Mesa.
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2.1 CONDITIONAL STATEMENTS
YOUR TURN. WRITE THE CONVERSE,
INVERSE, AND CONTRAPOSITIVE.
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA  20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA  20.
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2.1 CONDITIONAL STATEMENTS
REVIEW: LOGICAL STATEMENTS
Conditional:
If P, then Q.
Converse:
If Q, then P.
Inverse:
If not P, then not Q.
Contrapositive:
If not Q, then not P.
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2.1 CONDITIONAL STATEMENTS
DEFINITION: PERPENDICULAR LINES
Two lines that intersect to form a right angle.
n
Notation:
m
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mn
2.1 CONDITIONAL STATEMENTS
USING DEFINITIONS
You can write a definition as a conditional statement in if-then
form. Let’s look at an example:
Perpendicular Lines: two lines that intersect to form a right angle.
The conditional statement would be:
If two lines are perpendicular, then they intersect to form a
right angle.
The converse statement also ends up being true:
If two lines intersect to form a right angle, then they are
perpendicular lines.
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2.1 CONDITIONAL STATEMENTS
DAY 2
2.1 Conditional Statements
TRUTH VALUES
•A conditional is either True or False.
•To show that it is true, you must have an
argument (a proof) that it is true in all
cases.
•To show that it is false, you need to
provide at least one counterexample.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 7
True or false? If false provide a counter example.
If x2= 9, then x = 3.
FALSE!
Counterexample: x could be –3.
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2.1 CONDITIONAL STATEMENTS
EXAMPLE 8
If x = 10, then x + 4 = 14.
True!
Proof:
x = 10
x + 4 = 10 + 4
x + 4 = 14
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2.1 CONDITIONAL STATEMENTS
EQUIVALENT STATEMENTS
When two statements are both true or
both false, they are called equivalent
statements.
A conditional statement is always
equivalent to its contrapositive.
The inverse and converse are also
equivalent.
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2.1 CONDITIONAL STATEMENTS
EQUIVALENT STATEMENTS
Original:
TRUE
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA  20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA  20.
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False
False
TRUE
2.1 CONDITIONAL STATEMENTS
EXAMPLE 9
Statement: If x = 5, then x2 = 25.
TRUE
Contrapositive: If x2  25, then x  5. TRUE
Converse: If x2 = 25, then x = 5. FALSE – could be –5.
Inverse: If x  5, then x2  25.
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FALSE
2.1 CONDITIONAL STATEMENTS
JUSTIFYING STATEMENTS
In math, deciding if a statement is true or
false demands that you can justify your
answers. “Just because”, or, “It looks like
it” are not sufficient.
Justification must come in the form of
Postulates, Definitions, or Theorems.
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
Statement
D, X, and B are collinear.
EXAMPLE 10
A
D
X
B
Truth Value
TRUE
Reason
Definition of collinear
points.
C
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
Statement
EXAMPLE 11
AC  DB
Truth Value
TRUE
A
D
X
B
Reason
Definition of
Perpendicular lines
Def  lines
C
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
Statement
EXAMPLE 12
CXB is adjacent
to BXA
Truth Value
TRUE
A
Reason
D
X
B
Def. of adjacent angles
Def. of adj. s
C
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
Statement
DXA and CXB are
adjacent angles.
Truth Value
EXAMPLE 13
FALSE
A
D
X
B
Reason
There is not a common
side. (Or, they are
vertical angles.)
C
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
VERY IMPORTANT!
In doing proofs, you must be able to
justify every statement with a valid
reason. To be able to do this you must
know every definition, postulate and
theorem. Being able to look them up is no
substitute for memorization.
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
YOUR TURN
D
A
E B
F
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H
C
G
GEOMETRY 2.1 CONDITIONAL STATEMENTS
YOUR TURN
False (they are not collinear)
True (add to 180 )
True (post. 8)
D
A
E B
H
C
G
F
False (no rt.  mark)
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
YOUR TURN
True (def.  lines)
False (they are supplementary)
E B
True (half of 180 is 90 -- a right )
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D
A
H
C
F
G
GEOMETRY 2.1 CONDITIONAL STATEMENTS
BICONDITIONALS
When a conditional statement and its converse are both TRUE,
they can be written as a single biconditional statement. Let’s look
at an example:
Conditional
If 2 s are complementary, then their sum is 90°.
True
Converse
If the sum of 2 s is 90°, then they are complementary.
True
Biconditional
2 s are complementary if and only if their sum is 90°.
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2.1 CONDITIONAL STATEMENTS
BICONDITIONALS
(Continued)
Written with p’s and q’s a biconditional looks like this:
p if and only if q.
or
p iff q.
Iff means “if and only if”.
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2.1 CONDITIONAL STATEMENTS
PUTTING IT ALL TOGETHER
Statements
Conditional
Converse
Inverse
Contrapostive
Biconditional
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In words
In symbols
If p, then q
𝑝→𝑞
If q, then p
𝑞→𝑝
If not p, then not q
~𝑝 → ~𝑞
If not q, then not p
~𝑞 → ~𝑝
p if and only if q
𝑝↔𝑞
2.1 CONDITIONAL STATEMENTS
EXAMPLE 14
Let P be the statement: “x = 3”
Let Q be the statement: “2x = 6”
Write:
PQ
If x = 3, then 2x = 6.
QP
If 2x = 6, then x = 3.
PQ
x = 3 if and only if 2x = 6.
or 2x = 6 iff x = 3.
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2.3 DEDUCTIVE REASONING
51
DEFINITIONS
ALL definitions are biconditionals.
Example:
Definition of Congruent Angles
Two angles are congruent iff they have the same measure.
Conditional: If two angles are congruent, then they have the
same measure.
Converse: If two angles have the same measure, then they
are congruent.
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
TRUTH VALUES OF BICONDITIONALS
A biconditional is TRUE if both the
conditional and the converse are true.
A biconditional is FALSE if either the
conditional or the converse is false, or
both are false.
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
EXAMPLE 15
Biconditional
or False?
False!
x = 5 iff x2 = 25. True
Conditional
true or False?
If x = 5, then x2 = 25. True
Converse
False!or False?
If x2 = 25, then x = 5. True
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
YOUR TURN
Write the following biconditional statement as a
conditional statement and its converse.
An angle is obtuse iff it measures between 90 and 180.
Answer
Conditional: If an angle is obtuse, then it measures
between 90 and 180.
Converse: If an angle measures between 90 and 180,
then it is obtuse.
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GEOMETRY 2.1 CONDITIONAL STATEMENTS
WHY IS THIS IMPORTANT?
Geometry is stated in rules of logic.
We use logic to prove things.
It teaches us to think clearly and without error.
It impresses girl friends (or boy friends).
You can talk like…
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2.1 CONDITIONAL STATEMENTS
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2.1 CONDITIONAL STATEMENTS
ASSIGNMENT
2.1 DAY 1 #2-20 EVEN, 50, 64-68 EVEN
2.1 DAY 2 #26-36 EVEN, 37, 38, 46, 48
CHALLENGE PROBLEM #62
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2.1 CONDITIONAL STATEMENTS