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1
STANDARDS 1,2,3
What is a CONJECTURE?
CONDITIONALS: IF…, THEN….
CONVERSE OF A CONDITIONAL
END SHOW
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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2
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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3
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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4
STANDARDS 1,2,3
I will hit the target with this angle and pulling this way…
yes!
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5
STANDARDS 1,2,3
There it goes…!
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6
STANDARDS 1,2,3
Go ahead arrow…!
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7
STANDARDS 1,2,3
I didn’t think about the wind!
WIND
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8
STANDARDS 1,2,3
I didn’t think about the wind!
WIND
A CONJECTURE is an educated guess, and sometimes may be wrong.
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9
x
y
1
1
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)
K
L
N
M
Conjecture:
? They form a square!
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10
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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11
x
y
1
1
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)
A
B
D
C
Conjecture:
E
?!Or…
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12
x
y
1
1
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)
A
B
D
C
Conjecture:
Emhh!..or
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13
x
y
1
1
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)
A
B
D
C
Conjecture:
E
Guah! this also works…?
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14
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:
Sometimes we may reach to more than one conjecture!
x
y
A
B
D
C
E
?!?!
x
y
A
B
D
C
E?!x
y
A
B
D
C
E
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15
STANDARDS 1,2,3
Determine the validity of the conjecture and give a counterexample should the conjecture be false.
Given:
Conjecture:
A
B
D
CPoints A, B, C, D
They only form a square.
False!
Counterexample:
A
B
D
CThey could form an isosceles trapezoid as well!
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16
STANDARDS 1,2,3
CONDITIONAL STATEMENTS OR CONDITIONALS:
IF…, THEN …qp
If p, then q
p = hypothesis
q = conclusion
Where:
Students study to get good grades
If students study, then they get good grades.
If p, then q
Athletes train hard to win competitions.
If athletes train hard, then they win competitions.
Convert to conditional statements:
qp
HYPOTHESIS
HYPOTHESIS CONCLUSION
CONCLUSION
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17
STANDARDS 1,2,3
CONDITIONAL STATEMENTS OR CONDITIONALS:
IF…, THEN …qp
If p, then q
p = hypothesis
q = conclusion
Where:
CONVERSE:
IF…, THEN …pq
If q, then p
p = conclusion
q = hypothesis
Where:
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18
STANDARDS 1,2,3
Write the CONVERSE of the following conditional:
Athletes train hard to win competitions.
If athletes train hard, then they win competitions.
qp
HYPOTHESIS CONCLUSION
IF…, THEN …pq
CONVERSE:If they win competitions, then athletes train hard.
First convert to If…, then… statement
Now get the converse:
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19
STANDARDS 1,2,3
Write the CONVERSE of the following conditional:
IF…, THEN …pq
CONVERSE:If they get good grades, then students study.
First convert to If…, then… statement
Now get the converse:
Students study to get good grades
If students study, then they get good grades.
If p, then q
HYPOTHESIS CONCLUSION
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20
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 counterexamplePlan:
Write the given statement as a conditional:
If a linear pair, then angles are adjacent.
a) Converse: If angles are adjacent, then they are a linear pair.
b) It is false
c) Counterexample:
35°
55°Both angles in the figure at the right are adjacent but not a linear pair.
Solve:
The converse of a true conditional, not necessarily is true.PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
21
STANDARDS 1,2,3
NEGATION:The negation of a statement is its denial.
An angle is right An angle is not right
p ~p
~p is “not p” or the negation of p.
An angle is rightAn angle is not right
p ~p
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22
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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23
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.
b) Negating both the hypothesis and the conclusion:
If not a linear pair then it doesn’t have supplementary angles.
HYPOTHESIS
pCONCLUSION
q
Negated HYPOTHESIS
~pNegated CONCLUSION
~q
INVERSE
c) Is it true?
The inverse of this conditional is false, as shown in the following counterexample:
A
CB
D
E40°
140°In the figure at the left both angles ABC and EBD aren’t a linear pair but they are supplementary.
140° + 40° = 180°PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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 …qp
If p, then q
IF…, THEN …pq
If q, then p
CONVERSE: IF…, THEN …~p~q
If not q, then not p
CONTRAPOSITIVE:
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25
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.
FALSE. Line AB containing points A and B doesn’t lie in plane Q, but A and B do lie in plane R.
ABR
Q
Counterexample:
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26
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 numberp q
p q
4 and 6 are evenp
is true
is true
4 + 6 = 10, 10 is a real number.
q
Conclusion?
is true
By Law of Detachment
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27
STANDARDS 1,2,3Determine 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.
(b) Juan read a novel.
(c) He likes mystery books.
p qp q is true
p
q
is true
Yes, it follows by Law of Detachment.
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28
STANDARDS 1,2,3Determine 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 qp q is true
p
q
is true
Yes, it follows by Law of Detachment.
(b) m A + m B = 90°
(c) A and B are complementary
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29
STANDARDS 1,2,3Determine 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 qp q is true
p
q
is true
Invalid
(b) 1 and 2 are vertical.
(c) 1 and 2 oppose by the vertex.
What should follow to be true?
(c) 1 and 2 are congruent.
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30
STANDARDS 1,2,3
LAW OF SYLLOGISM:
If p q and q r are true conditionals, then p r is true as well.
If is true, then is true.
p q
q rp r
If the vehicle has four wheels,
then you can drive it.
p
q
q
r
p
r
If a vehicle has four wheels, then it is a car
If it is a car, then you can drive it.
Using the Law of Syllogism, what conclusion may be reached?
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31
STANDARDS 1,2,3
(c) If a mammal, then it drinks milk.
p
q
q
r
(a) If a mammal, then it has warm blood.
(b) If it has warm blood then it drinks milk.
Determine if statement (c) follows from statements (a) and (b) by the Law of Syllogism. In case this is not true, write INVALID.
p r
Yes, by the Law of Syllogism.
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32
STANDARDS 1,2,3
p
q
q
r
Determine if statement (c) follows from statements (a) and (b) by the Law of Syllogism. In case this is not true, write INVALID.
Yes, by the Law of Syllogism.
(b) ABC is a right angle.
(c) A is a right angle.
(a) A ABC
p r
If ABC, then it is a right angle.
If A, then it is a right angle.
If A, then congruent to ABC p
q
q
r
p r
Each statement could be read as:
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33
STANDARDS 1,2,3
p
q
q
r
Can a conclusion be reached using the Law of Detachment or the Law of Syllogism from (a) and (b)
(b) An obtuse angle is greater than an acute angle.
(a) ABC is an obtuse angle.
p r ABC is greater than an acute angle
by Law of Syllogism
CONCLUSION:
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Logical Reasoning
- Uses a set of rules to prove a statement.
- Finding a general rule based on observation of data, patterns,and past performance.
Deductive Reasoning Inductive Reasoning
4x + 2 = 22Given:
Prove:
x = 5
4x + 2 = 22
-2 -2
4x = 20
Subtraction Property of Equality
Proof:
4
53
32
11
SquaresStep
7
Rule: We add 2 squares per step.
?4 4
x = 5
Division Property of Equality
Substitution Property of Equality
STANDARDS 1,2,3
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35
PROPERTIES OF REAL NUMBERS
COMMUTATIVE PROPERTY:
Addition: a + b = b + a 5 + 7 = 7 + 5
Multiplication: 9 6 = 6 9
For any real numbers a, b, and c:
a b = b a
ALGEBRAIC REVIEWSTANDARDS 1,2,3
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36
PROPERTIES OF REAL NUMBERS
ASSOCIATIVE PROPERTY:
Addition: (a + b) + c = a + (b + c)
(3 + 4) +1 = 3 + (4 + 1)
Multiplication:
For any real numbers a, b, and c:
34 45 6 = 34 45 6a b c= a b c
STANDARDS 1,2,3
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37
PROPERTIES OF REAL NUMBERS
IDENTITY PROPERTY:
Addition: a + 0 = 0 + a=a 5 + 0 = 0 + 5
Multiplication: 9 1 = 1 9
For any real numbers a, b, and c:
a 1 = 1 a = a
= 9
= 5
STANDARDS 1,2,3
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38
PROPERTIES OF REAL NUMBERS
INVERSE PROPERTY:
Addition: a + (-a) = (-a) + a=0 5 + (-5) = (-5) + 5
Multiplication:
For any real numbers a, b, and c:
= 1
= 1
= 0
a = a = 1 1a
1a
If a=0 then35
53
15
5
=53
35
15
= 5
STANDARDS 1,2,3
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39
PROPERTIES OF REAL NUMBERS
DISTRIBUTIVE PROPERTY:
Distributive:
For any real numbers a, b, and c:
a(b+c) = ab + ac (b+c)a = ba + caand
3(5+1) = 3(5) + 3(1) (5+1)3 = 5(3) + 1(3)and
STANDARDS 1,2,3
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40
Name the property shown at each equation:
1 45 = 45a)
56 + 34 = 34 + 56b)
(-3) + 3 = 0c)
5(9 +2) = 45 + 10d)
(2 + 1) +b= 2 + (1 + b)e)
-34(23) = 23(-34)f)
Identity property (X)
Commutative property (+)
Inverse property (+)
Distributive property
Associative property (+)
Commutative property (X)
STANDARDS 1,2,3
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41
ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY:
PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW
For any numbers a, b, and c, if a=b then a+c=b+c and a-c=b-c
10 = 10+ 6 +616 = 16
22 = 22-5 -5 17 = 17
STANDARDS 1,2,3
SUBSTITUTION PROPERTY OF EQUALITY:
If a=b, then a may be replaced by b. b=2 and 3b +1=7If
then 3( )+1=72
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42
MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY:
PROPERTIES OF EQUALITY
For any real numbers a, b, and c, if a=b, then a c=b c and if c=0, =ac
bc
15 = 152 230 = 30
28 = 287 74 = 4
24 = 243 372 = 72
36 = 3612 123 = 3
STANDARDS 1,2,3
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Deductive Reasoning: AlgebraSTANDARDS 1,2,3
Given: 4(x + 2) = 2x + 18
Prove: x = 5
(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.
Two column proofs:
Proof:
Statements Reasons
FORMAL INFORMAL
4(x + 2) = 2x + 18
4x + 8= 2x + 18
4x = 2x + 10
2x = 10
x = 5
-8 -8
-2x -2x
2 2
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44
Deductive Reasoning (GEOMETRY)
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.
STANDARDS 1,2,3
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45
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 5=X
6X-12=8 8=6X-12
9Y -2Y +1= 3X2 3X= 9Y -2Y+12
STANDARDS 1,2,3
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=6-3XPRESENTATION CREATED BY SIMON PEREZ. All rights reserved
46
STANDARDS 1,2,3
of segments is transitive. of s is transitive
of segments is symmetric. of s is symmetric
of segments is reflexive. of s is reflexive
KL LM
LM AB
KL AB
KL LM LM KL
LMLM
BCE FGH
FGH ECA
BCE ECA
BCE FGH BCEFGH
ECA ECA
Congruence in segments and angles is Reflexive, Symmetric and Transitive:
For all segments and angles, their measures comply with these same properties.PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
47
STANDARDS 1,2,3DEDUCTIVE REASONING: GEOMETRY (formal)
Two Column Proof:
Statements Reasons
(1) (1) Given
(2) (2)
(3) (3)
(4) (4)
L is midpoint of KM
KL LM Definition of Midpoint
LM AB Given
KL AB of segments is transitive.
Given:
Prove:
MLK
BA
L is midpoint of KM
LM AB
KL AB
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48
B
CF
D
A
E
STANDARDS 1,2,3
Given:
EFD is right
Prove:
AFB CFBand are complementary.
DEDUCTIVE REASONING: GEOMETRY (formal)
Two Column Proof:
Statements Reasons
(1) (1)EFD is right Given
(2) (2)EC AD Definition of lines
(3) (3)AFC is right lines form 4 right s
(4) (4)AFC=m 90° Definition of right s
(5) (5)AFB +m AFCmCFB =m
(6) (6)
addition postulate
AFB +m CFB = 90°m Substitution prop. of (=)
AFB CFBand are complementary.
(7) (7) Definition of complementary s
B
CF
D
A
E
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49
STANDARDS 1,2,3
Given:
Prove:
DEDUCTIVE REASONING: GEOMETRY (formal)
Two Column Proof:
Statements Reasons
(1) (1)
(2) (2)
(3) (3)
(4) (4)
(5) (5)
(6) (6)
(7) (7)
(8) (8)
(9) (9)
B
ACE
D
F
GH
CE bisects BCA
FGH ECA
FGH +m BCD = 180°m2( )
CE bisects BCA Given
BCE ECA Definition of bisector
BCE=m ECAm Definition of s
FGH ECA Given
FGH=m ECAm
BCE=m FGHm
Definition of s
of s is transitive
BCE +m BCD = 180°mECA +m addition postulate
FGH +m BCD = 180°mFGH +m
FGH +m BCD = 180°m2( )
Substitution prop. of (=)
Adding like terms
B
ACE
D
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50
STANDARDS 1,2,3
Given:
FBD is right
Prove:
ABF CBDand are complementary.
DEDUCTIVE REASONING: GEOMETRY (formal)
Two Column Proof:
Statements Reasons
(1) (1)
(2) (2)
(3) (3)
(4) (4)
(5) (5)
(6) (6)
F
EB
C
A
D
F
EB
C
A
D
FBD is right Given
FBD=m 90° Definition of right s
FBD +m CBD = 180°mABF +m addition postulate
CBD = 180°mABF +m 90° +
ABF +m CBD = 90°m
Substitution prop. of (=)
Subtraction prop. of (=)
ABF CBDand are complementary. Definition of complementary s
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51
STANDARDS 1,2,3
Given:
Prove:
DEDUCTIVE REASONING: GEOMETRY (formal)
Two Column Proof:
Statements Reasons
(1) (1)
(2) (2)
(3) (3)
(4) (4)
(5) (5)
(6) (6)
(7) (7)
CAB
H
G
D E F
CAB
H
G
D E FGE is a transversalAC and DF are
GBC FEHand are supplementary.
GE is a transversal
AC and DF are Given
GBC CBEand are a linear pair Definition of linear pair
GBC +m CBE = 180°m s in a linear pair are supplementary
CBE FEH In lines cut by a transversal CORRESPONDING s are
Definition of sCBE=m FEHm
GBC +m FEH = 180°m Substitution prop. of (=)
GBC FEHand are supplementary.
Definition of supplementary s
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