Transcript STANDARDS 1,2,3 What is a CONJECTURE? CONDITIONALS: IF…, THEN…. CONVERSE OF A CONDITIONAL NEGATION OF A CONDITIONAL INVERSE OF A CONDITIONAL CONTRAPOSITIVE OF A CONDITIONAL LAW OF.

STANDARDS 1,2,3
What is a CONJECTURE?
CONDITIONALS: IF…, THEN….
CONVERSE OF A CONDITIONAL
NEGATION OF A CONDITIONAL
INVERSE OF A CONDITIONAL
CONTRAPOSITIVE OF A CONDITIONAL
LAW OF DETACHMENT
LAW OF SYLLOGISM
DEDUCTIVE VS INDUCTIVE?
ELEMENTS TO CONSTRUCT PROOFS
GEOMETRIC PROOF 1
GEOMETRIC PROOF 2
GEOMETRIC PROOF 3
GEOMETRIC PROOF 4
GEOMETRIC PROOF 5
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END SHOW
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Standard 1:
Students demonstrate understanding by identifying
and giving examples of undefined terms, axioms,
theorems, and inductive and deductive reasoning.
Standard 2:
Students write geometric proofs, including proofs
by contradiction.
Standard 3:
Students construct and judge the validity of a
logical argument and give counterexamples to
disprove a statement.
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Estándar 1:
Los estudiantes demuestran entendimiento en
identificar ejemplos de términos indefinidos,
axiomas, teoremas, y razonamientos inductivos y
deductivos.
Standard 2:
Los estudiantes escriben pruebas geométricas,
incluyendo pruebas por contradicción.
Standard 3:
Los estudiantes construyen y juzgan la validéz de
argumentos lógicos y dan contra ejemplos para
desaprobar un estatuto.
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STANDARDS 1,2,3
I will hit the target with this
angle and pulling this
way…yes!
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STANDARDS 1,2,3
There it goes…!
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STANDARDS 1,2,3
Go ahead arrow…!
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STANDARDS 1,2,3
WIND
I didn’t think about
the wind!
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STANDARDS 1,2,3
WIND
I didn’t think about
the wind!
A CONJECTURE is an educated guess,
and sometimes may be wrong.
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STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: K(1,1), L(1,3), M(3,3), N(3,1)
y
Conjecture:
?
They form a square!
L
M
K
N
1
1
x
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STANDARDS 1,2,3
What conjecture may be made from the given information?
Given:
A
B
D
E
Conjecture:
Point E noncollinear with points A, B, and D.
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STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2)
Conjecture:
?!
y
B
Or…
C
E
1
A
1
D
x
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STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2)
y
Conjecture:
B
mhh!..or
C
E
1
A
1
D
x
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STANDARDS 1,2,3
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2)
y
Conjecture:
Guah! this also works…?
B
C
E
1
A
1
D
x
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STANDARDS 1,2,3
y
What conjecture may be made from the given information?
Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2)
C
B
y
E
Conjecture:
B
?!
?!
A
C
D
x
E
A
D
x
y
?!
B
C
E
A
Sometimes we may reach to more than one conjecture!
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D
x
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STANDARDS 1,2,3
Determine the validity of the conjecture and give a counterexample
should the conjecture be false.
Given:
Points A, B, C, D
B
C
A
D
Conjecture:
They only form a square.
False!
Counterexample:
B
A
They could form an isosceles
trapezoid as well!
C
D
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STANDARDS 1,2,3
CONDITIONAL STATEMENTS OR CONDITIONALS:
IF…, THEN …
p
q
Where:
p = hypothesis
q = conclusion
If p, then q
Convert to conditional statements:
HYPOTHESIS
CONCLUSION
Students study to get good grades
If p, then q
If students study, then they get good grades.
HYPOTHESIS
CONCLUSION
Athletes train hard to win competitions.
p
q
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If athletes train hard, then they win competitions.
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STANDARDS 1,2,3
CONDITIONAL STATEMENTS OR CONDITIONALS:
IF…, THEN …
p
q
Where:
p = hypothesis
q = conclusion
If p, then q
CONVERSE:
IF…, THEN …
q
p
Where:
p = conclusion
q = hypothesis
If q, then p
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STANDARDS 1,2,3
Write the CONVERSE of the following conditional:
HYPOTHESIS
CONCLUSION
Athletes train hard to win competitions.
p
q
First convert to If…, then…
statement
If athletes train hard, then they win competitions.
Now get the converse:
If they win competitions, then athletes train hard.
CONVERSE:
IF…, THEN …
q
p
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STANDARDS 1,2,3
Write the CONVERSE of the following conditional:
HYPOTHESIS
CONCLUSION
Students study to get good grades
If p, then q
If students study, then they get good grades.
First convert to If…, then…
statement
Now get the converse:
If they get good grades, then students study.
CONVERSE:
IF…, THEN …
q
p
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STANDARDS 1,2,3
Write the converse of the following true statement, and determine if true or
false. If it is false, give a counterexample:
A linear pair has adjacent angles.
Explore:
a) Obtain converse
b) Is it true or false?
c) If false find a counterexample
Plan:
Write the given statement as a conditional:
If a linear pair, then angles are adjacent.
Solve:
a) Converse: If angles are adjacent, then they are a linear pair.
b) It is false
c) Counterexample:
Both angles in the figure at the right
are adjacent but not a linear pair.
The converse of a true conditional, not necessarily is true.
55°
35°
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STANDARDS 1,2,3
NEGATION:
The negation of a statement is its denial.
~p is “not p” or the negation of p.
An angle is right
p
An angle is not right
~p
An angle is not right
An angle is right
p
~p
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STANDARDS 1,2,3
INVERSE:
The inverse of a conditional statement is when both the hypothesis and the
conclusion are denied.
~p
~q
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STANDARDS 1,2,3
For the true conditional: a linear pair has supplementary angles; write the
inverse and determine if true or false. If false give a counterexample:
a)Writing the conditional in If-Then form:
If a linear pair, then it has supplementary angles.
HYPOTHESIS
CONCLUSION
p
q
b) Negating both the hypothesis and the conclusion:
If not a linear pair then it doesn’t have supplementary angles.
INVERSE
Negated HYPOTHESIS
Negated CONCLUSION
~p
~q
c) Is it true?
The inverse of this conditional is false, as shown in the following counterexample:
A
40°
140°
B C
D
E
In the figure at the left both angles ABC
and EBD aren’t a linear pair but they are
supplementary.
140° + 40° = 180°
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STANDARDS 1,2,3
CONTRAPOSITIVE of a conditional statement:
The contrapositive of a conditional statement is the negation of the
hypothesis and conclusion of its converse.
IF…, THEN …
p
q
If p, then q
CONTRAPOSITIVE:
CONVERSE:
IF…, THEN …
q
p
If q, then p
IF…, THEN …
~q
~p
If not q, then not p
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STANDARDS 1,2,3
Find the contrapositive of the true conditional if two points lie in a plane, then
the entire line containing those points lies in that plane. Is the contrapositive
true or false?
a) converse:
If the entire line containing those points lies in that plane, then the
two points lie in a plane.
b) contrapositive:
If the entire line containing those points does not lie in that plane,
then the two points do not lie in a plane.
Counterexample:
FALSE.
Line AB containing points A and B doesn’t lie in
plane Q, but A and B do lie in plane R.
R
A
B
Q
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STANDARDS 1,2,3
LAW OF DETACHMENT
If
p q is a true statement and p is true, then q is true.
If two numbers are even, then their sum is a real number is a true conditional, and
4 and 6 are even numbers. Try to reach a logical conclusion using the Law of
Detachment.
If two numbers are even, then their sum is a real number
q
p
p q is true
4 and 6 are even
p
is true
Conclusion?
4 + 6 = 10, 10 is a real number.
q
is true
By Law of Detachment
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STANDARDS 1,2,3
Determine if statement (c) goes after statements (a) and (b) by the Law of
Detachment. If it does not follow, then write invalid [suppose (a) and (b)
true]:
(a) If you read novels, then you like mystery books.
p q is true
p
q
(b) Juan read a novel.
p
is true
(c) He likes mystery books.
Yes, it follows by Law of Detachment.
q
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STANDARDS 1,2,3
Determine if statement (c) goes after statements (a) and (b) by the Law of
Detachment. If it does not follow, then write invalid [suppose (a) and (b)
true]:
(a) If two angles add up to 90° then they are complementary p q
is true
p
q
(b) m A + m B = 90°
p
(c)
A and
is true
B are complementary
q
Yes, it follows by Law of Detachment.
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STANDARDS 1,2,3
Determine if statement (c) goes after statements (a) and (b) by the Law of
Detachment. If it does not follow, then write invalid [suppose (a) and (b)
true]:
(a) If two angles are vertical, then they are congruent
p q is true
p
q
(b)
1 and
(c)
1 and
2 are vertical.
p
is true
2 oppose by the vertex.
q
Invalid
What should follow to be true?
(c)
1 and
2 are congruent.
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STANDARDS 1,2,3
LAW OF SYLLOGISM:
If p
q and q
p
r are true conditionals, then p
q
is true, then
If
q
r is true as well.
p
r
is true.
r
Using the Law of Syllogism, what conclusion may be reached?
If a vehicle has four wheels, then it is a car
p
If the vehicle has four wheels,
If it is a car, then you can drive it.
q
p
q
r
then you can drive it.
r
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STANDARDS 1,2,3
Determine if statement (c) follows from statements (a) and (b) by the Law of
Syllogism. In case this is not true, write INVALID.
(a) If a mammal, then it has warm blood.
p
q
(b) If it has warm blood then it drinks milk.
q
(c)
r
If a mammal, then it drinks milk.
p
r
Yes, by the Law of Syllogism.
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STANDARDS 1,2,3
Determine if statement (c) follows from statements (a) and (b) by the Law of
Syllogism. In case this is not true, write INVALID.
Each statement could be read as:
(a)
A
p
(b)
ABC
If
q
p
ABC is a right angle.
r
q
(c)
A, then congruent to
If
ABC
q
ABC, then it is a right angle.
q
r
A is a right angle.
p
r
If
A, then it is a right angle.
p
r
Yes, by the Law of Syllogism.
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STANDARDS 1,2,3
Can a conclusion be reached using the Law of Detachment or the Law of
Syllogism from (a) and (b)
(a)
ABC is an obtuse angle.
p
q
(b) An obtuse angle is greater than an acute angle.
q
r
CONCLUSION:
ABC is greater than an acute angle
p
r
by Law of Syllogism
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Logical Reasoning
Deductive Reasoning
Inductive Reasoning
- Uses a set of rules to prove a
statement.
Given:
4x + 2 = 22
Prove:
x=5
Proof:
4x + 2 = 22
-2
-2
- Finding a general rule
based on observation of
data, patterns,
and past performance.
Subtraction Property of
Equality
4x = 20
4 4
Division Property of
Equality
x=5
Substitution Property
of Equality
STANDARDS 1,2,3
Step
Squares
1
1
2
3
3
5
4
7
Rule: We add 2 squares per step.
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?
ALGEBRAIC REVIEW
STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
COMMUTATIVE PROPERTY:
Addition:
a+b=b+a
Multiplication: a b = b a
5+7 =7+5
9 6 =6 9
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STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
ASSOCIATIVE PROPERTY:
Addition:
(a + b) + c = a + (b + c)
Multiplication: a b c= a b c
(3 + 4) +1 = 3 + (4 + 1)
34 45 6 = 34 45 6
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STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
IDENTITY PROPERTY:
Addition:
a + 0 = 0 + a=a
Multiplication: a 1 = 1 a = a
5+0 =0+5 =5
9 1 =1 9 =9
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STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
INVERSE PROPERTY:
Addition: a + (-a) = (-a) + a=0
5 + (-5) = (-5) + 5 = 0
If a=0 then
Multiplication:
a 1= 1 a = 1
a a
1= 1
5
5 5
5 3 3
=
3 5 5
5=1
5
=1
3
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STANDARDS 1,2,3
PROPERTIES OF REAL NUMBERS
For any real numbers a, b, and c:
DISTRIBUTIVE PROPERTY:
Distributive:
a(b+c) = ab + ac
3(5+1) = 3(5) + 3(1)
and
and
(b+c)a = ba + ca
(5+1)3 = 5(3) + 1(3)
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STANDARDS 1,2,3
Name the property shown at each equation:
a) 1 45 = 45
Identity property (X)
b) 56 + 34 = 34 + 56
Commutative property (+)
c) (-3) + 3 = 0
Inverse property (+)
d) 5(9 +2) = 45 + 10
Distributive property
e) (2 + 1) +b= 2 + (1 + b)
Associative property (+)
f) -34(23) = 23(-34)
Commutative property (X)
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STANDARDS 1,2,3
PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
SUBSTITUTION PROPERTY OF EQUALITY:
If a=b, then a may be replaced by b.
If b=2 and 3b +1=7
then
3( 2 )+1=7
ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY:
For any numbers a, b, and c, if a=b then a+c=b+c and a-c=b-c
10 = 10
+ 6 +6
16 = 16
22 = 22
-5 -5
17 = 17
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STANDARDS 1,2,3
PROPERTIES OF EQUALITY
MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY:
a b
For any real numbers a, b, and c, if a=b, then a c=b c and if c=0, c = c
2 15 = 15 2
30 = 30
28 = 28
7
7
4=4
3 24 = 24 3
72 = 72
36 = 36
12 12
3=3
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Deductive Reasoning: Algebra
STANDARDS 1,2,3
INFORMAL
FORMAL
Two column proofs:
Given: 4(x + 2) = 2x + 18
Prove: x = 5
Proof:
Statements
Reasons
(1) 4(x + 2) = 2x + 18
(1) given
(2) 4x + 8= 2x + 18
(2) Distributive prop.
(3) 4x = 2x + 10
(3) Subtraction prop. of equality
(4) 2x = 10
(4) Subtraction prop. of equality
(5) x = 5
(5) Division Prop. of equality.
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4(x + 2) = 2x + 18
4x + 8= 2x + 18
-8
-8
4x = 2x + 10
-2x -2x
2x = 10
2
2
x=5
Deductive Reasoning (GEOMETRY)
STANDARDS 1,2,3
Conjecture - a statement or conditional trying to prove.
Elements to construct proofs:
a) Undefined terms - Terms that are so obvious that don’t require to
be proven.
point, line, etc.
b) Definitions - Statements defined using other terms.
Triangle is a 3 sided polygon.
c) Axioms (Postulates) - Statements or properties that don’t need to
be proven to be used in proofs.
If two planes intersect their intersection is a line.
d) Theorems - Statements or properties that require to be proven
to be used in proofs.
If two angles form a linear pair, then they are supplementary angles.
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44
STANDARDS 1,2,3
PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
REFLEXIVE PROPERTY OF EQUALITY:
For any real number a, a=a
5=5
-10=-10
SYMMETRIC PROPERTY OF EQUALITY:
For all real numbers a and b, if a=b, then b=a
X=5
6X-12=8
9Y 2-2Y +1= 3X
5=X
8=6X-12
3X= 9Y 2-2Y+1
TRANSITIVE PROPERTY OF EQUALITY:
For all real numbers a, b, and c, if a=b, and b=c then a=c
If X=6 and Y= 6 then X=Y
If Y=2X+2 and Y=6-3X then 2X+2=6-3X
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45
STANDARDS 1,2,3
Congruence in segments and angles is Reflexive, Symmetric and Transitive:
of segments is reflexive.
LM
of
ECA
LM
of segments is symmetric.
KL
LM
LM
s is reflexive
KL
of
BCE
of segments is transitive.
s is symmetric
FGH
of
ECA
FGH
s is transitive
AB
BCE
FGH
FGH
ECA
AB
BCE
ECA
KL
LM
LM
KL
BCE
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For all segments and angles, their measures comply with these same properties.
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DEDUCTIVE REASONING: GEOMETRY (formal)
Given:
L is midpoint of KM
K
LM
AB
Prove:
KL AB
Two Column Proof:
(3) LM
(4) KL
AB
AB
M
A
B
Reasons
Statements
(1) L is midpoint of KM
(2) KL
LM
L
STANDARDS 1,2,3
(1) Given
(2) Definition of Midpoint
(3) Given
(4)
of segments is transitive.
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STANDARDS 1,2,3
DEDUCTIVE REASONING: GEOMETRY (formal)
B
A
Given:
EFD is right
F
E
C
F
E
D
Prove:
AFB and
B
A
C
D
CFB are complementary.
Two Column Proof:
Reasons
Statements
(1)
(1) Given
(2) Definition of
EFD is right
(2) EC AD
(3) AFC is right
(3)
(4) m AFC= 90°
(5) m AFB + m CFB = m
(6) m
(7)
AFB + m
CFB = 90°
AFB and CFB are
complementary.
lines
(4) Definition of right
AFC (5)
s
lines form 4 right
s
addition postulate
(6) Substitution prop. of (=)
(7) Definition of
complementary
s
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STANDARDS 1,2,3
DEDUCTIVE REASONING: GEOMETRY (formal)
Given:
B
F
CE bisects BCA
E
H
C A
FGH
ECA
D
G
Prove:
2( m FGH) + m
B
E
D
C
BCD = 180°
Two Column Proof:
Statements
Reasons
BCA
(1) Given
(2) Definition of
bisector
(3) Definition of
s
(1) CE bisects
(2)
BCE
ECA
(3) m BCE= m ECA
(4)
FGH
(5) m
(6) m
(7) m
(8) m
(9) 2( m
(4) Given
ECA
FGH=m
BCE= m
ECA + m
FGH + m
ECA
(5) Definition of
FGH
(6)
BCE + m
FGH + m
FGH) + m
BCD = 180°
BCD = 180°
BCD = 180°
of
s
s is transitive
addition postulate
(7)
(8) Substitution prop. of (=)
(9) Adding like terms
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49
A
STANDARDS 1,2,3
DEDUCTIVE REASONING: GEOMETRY (formal)
A
Given:
FBD is right
F
B
A
B
E
C D
Prove:
ABF and
F
E
C D
CBD are complementary.
Two Column Proof:
Reasons
Statements
FBD is right
(2) m FBD= 90°
(3) m ABF + m FBD + m
(1) Given
(1)
(4) m
ABF +
(5) m
ABF + m
(6)
ABF and
90° +
m
(2) Definition of right s
CBD = 180° (3)
addition postulate
CBD = 180° (4) Substitution prop. of (=)
CBD = 90°
CBD are complementary.
(5) Subtraction prop. of (=)
(6) Definition of
complementary
s
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STANDARDS 1,2,3
DEDUCTIVE REASONING: GEOMETRY (formal)
Given:
G
C
A
AC and DF are
B
GE is a transversal
D E
F
H
Prove:
GBC and FEH are supplementary.
G
A
B
D
H
E
F
Two Column Proof:
Reasons
Statements
AC and DF are
GE is a transversal
(2)
GBC and CBE are a linear pair
(1)
(3) m GBC + m CBE = 180°
(4)
CBE
FEH
(5) m
CBE= m
(6) m
GBC + m
(7)
FEH
FEH = 180°
GBC and FEH are
supplementary.
(1) Given
(2) Definition of linear pair
(3) s in a linear pair are supplementary
(4) In lines cut by a transversal
CORRESPONDING
s are
(5) Definition of
s
(6) Substitution prop. of (=)
(7) Definition of
supplementary
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C
s
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