Geometry - Mesa Public Schools · September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS In math,...
-
Upload
phungtuyen -
Category
Documents
-
view
215 -
download
0
Transcript of Geometry - Mesa Public Schools · September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS In math,...
ESSENTIAL QUESTION
When is a conditional statement true or false?
September 7, 2016 2.1 CONDITIONAL STATEMENTS
WHAT YOU WILL LEARN
oWrite conditional statements.
oUse definitions written as conditional statements.
oWrite biconditional statements.
September 7, 2016 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.
September 7, 2016 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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
EXAMPLE 1
If I study hard, then I will get good grades.
HYPOTHESIS
I study hard
CONCLUSION
I will get good grades.
September 7, 2016 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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
YOUR TURN
Underline the hypothesis and circle the conclusion.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
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.
REWRITING STATEMENTS.
oUse common sense.
oDon’t over analyze it.
oMake sure the sentence is grammatically correct.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
The hypothesis always follows “IF.”
No “if?” The first part is usually the hypothesis.
Make your English teacher proud!Does it sound right?
EXAMPLE 2A
Rewrite the following statement in if-then form:
All birds have feathers.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
What is the hypothesis?
What is the conclusion? have feathers
All birds
If-then form?
If an animal is a bird, then it has feathers.
EXAMPLE 2B
Rewrite the following statement in if-then form:
You are in Texas if you are in Houston.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
What is the hypothesis?
What is the conclusion? You are in Texas
You are in Houston
If-then form?
If you are in Houston, then you are in
Texas.
EXAMPLE 2C
Rewrite the following statement in if-then form:
An even number is divisible by 2.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
What is the hypothesis?
What is the conclusion? Divisible by 2.
An even number
If-then form?
If a number is even, then it is divisible by 2.
YOUR TURN
Rewrite the conditional statement in if-then form.
If yesterday was Sunday, then today is
Monday.
If an object measures 12 inches, then it is one
foot long.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
3. Today is Monday if yesterday was Sunday.
4. An object that measures 12 inches is one foot long.
NEGATION
The negative of the original statement. Examples:
I am happy.
I am not happy.
mC = 30°.
mC 30°.
A, B and C are on the same line.
A, B and C are not on the same line.
September 7, 2016 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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
RELATED CONDITIONAL STATEMENTS
Looking at the conditional statement: If p, then q.
There are three similar statements we can make.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
o Converseo Inverseo Contrapositive
CONVERSE
The converse of a statement is formed by
switching the hypothesis and the conclusion.
If you play drums, then you are in the band.
Conditional:
Converse:
If you are in the band, then you play drums.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
If Q, then P.
EXAMPLE 4
Write the converse of the statement below.
Answer:
If you play on the tennis team, then you like tennis.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
If you like tennis, then you play on the tennis team.
INVERSE
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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
If not P, then not Q.
EXAMPLE 5
Write the inverse of the statement below.
Answer:
If today is not Monday, then tomorrow is not Tuesday.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
If today is Monday, then tomorrow is Tuesday.
CONTRAPOSITIVE
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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
If not Q, then not P.
EXAMPLE 6
Write the contrapositive of the statement below.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
If x is odd, then x + 1 is even.
x + 1 is not evenNegateNegate
x is not odd
If x+1 is not even, then x is not odd.
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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
YOUR TURN. WRITE THE CONVERSE, INVERSE, AND CONTRAPOSITIVE.
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA 20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA 20.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
REVIEW: LOGICAL STATEMENTS
September 7, 2016 2.1 CONDITIONAL 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.
DEFINITION: PERPENDICULAR LINES
September 7, 2016 2.1 CONDITIONAL STATEMENTS
Two lines that intersect to form a right angle.
m
n
Notation:
m n
USING DEFINITIONS
You can write a definition as a conditional statement in if-then form. Let’s look at an example:
The conditional statement would be:
The converse statement also ends up being true:
September 7, 2016 2.1 CONDITIONAL STATEMENTS
Perpendicular Lines: two lines that intersect to form a right angle.
If two lines are perpendicular, then they intersect to form a
right angle.
If two lines intersect to form a right angle, then they are
perpendicular lines.
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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
EXAMPLE 7True or false? If false provide a counter example.
If x2= 9, then x = 3.
FALSE!Counterexample: x could be –3.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
EXAMPLE 8
If x = 10, then x + 4 = 14.
True! Proof:
x = 10
x + 4 = 10 + 4
x + 4 = 14
September 7, 2016 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.
September 7, 2016 2.1 CONDITIONAL STATEMENTS
EQUIVALENT STATEMENTS
Original:
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA 20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA 20.
TRUE
False
False
TRUE
September 7, 2016 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. FALSE
September 7, 2016 2.1 CONDITIONAL STATEMENTS
JUSTIFYING STATEMENTS
September 7, 2016 GEOMETRY 2.1 CONDITIONAL 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.
EXAMPLE 10
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
A
XD B
C
Statement
Truth Value
Reason
D, X, and B are collinear.
TRUE
Definition of collinear points.
EXAMPLE 11
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
A
XD B
C
Statement
Truth Value
Reason
AC DB
TRUE
Definition of Perpendicular lines
Def lines
EXAMPLE 12
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
A
XD B
C
Statement
Truth Value
Reason
CXB is adjacent to BXA
TRUE
Def. of adjacent angles
Def. of adj. s
EXAMPLE 13
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
A
XD B
C
Statement
Truth Value
Reason
DXA and CXB are adjacent angles.
FALSE
There is not a common side. (Or, they are vertical angles.)
VERY IMPORTANT!
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
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.
YOUR TURN
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
A
F
BE
D
C
G
H
False (they are not collinear)
True (add to 180 )
True (post. 8)
False (no rt. mark)
YOUR TURN
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
A
F
BE
D
C
G
H
True (def. lines)
False (they are supplementary)
True (half of 180 is 90 -- a right )
BICONDITIONALS
September 7, 2016 2.1 CONDITIONAL STATEMENTS
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
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
Biconditional
2 s are complementary if and only if their sum is 90°.
BICONDITIONALS (Continued)
September 7, 2016 2.1 CONDITIONAL STATEMENTS
Written with p’s and q’s a biconditional looks like this:
p if and only if q.
p iff q. or
Iff means “if and only if”.
PUTTING IT ALL TOGETHER
September 7, 2016 2.1 CONDITIONAL STATEMENTS
Statements In words In symbols
Conditional If p, then q 𝑝 → 𝑞
Converse If q, then p 𝑞 → 𝑝
Inverse If not p, then not q ~𝑝 → ~𝑞
Contrapostive If not q, then not p ~𝑞 → ~𝑝
Biconditional p if and only if q 𝑝 ↔ 𝑞
EXAMPLE 14
September 7, 2016 2.3 DEDUCTIVE REASONING 51
Let P be the statement: “x = 3”
Let Q be the statement: “2x = 6”
Write:
P Q
Q P
P Q
If x = 3, then 2x = 6.
If 2x = 6, then x = 3.
x = 3 if and only if 2x = 6.
or 2x = 6 iff x = 3.
DEFINITIONS
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
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.
TRUTH VALUES OF BICONDITIONALS
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
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.
EXAMPLE 15
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
Biconditional
x = 5 iff x2 = 25.
Conditional
If x = 5, then x2 = 25.
Converse
If x2 = 25, then x = 5.
true
False!
False!
True or False?
True or False?
True or False?
YOUR TURN
September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS
Write the following biconditional statement as a conditional statement and its converse.
An angle is obtuse iff it measures between 90 and 180.
AnswerConditional: 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.
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…
September 7, 2016 2.1 CONDITIONAL STATEMENTS