1 standards 1,2,3 what is a conjecture? conditionals: if…, then…. converse of a conditional end...

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

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

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.



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.



4

STANDARDS 1,2,3

I will hit the target with this angle and pulling this way…

yes!



5

STANDARDS 1,2,3

There it goes…!



6

STANDARDS 1,2,3

Go ahead arrow…!



7

STANDARDS 1,2,3

I didn’t think about the wind!

WIND



8

STANDARDS 1,2,3

I didn’t think about the wind!

WIND

A CONJECTURE is an educated guess, and sometimes may be wrong.



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!



10

STANDARDS 1,2,3


Given:

A B D

E

Conjecture:

Point E noncollinear with points A, B, and D.



11

x

y

1

1

STANDARDS 1,2,3


Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2)

A

B

D

C

Conjecture:

E

?!Or…



12

x

y

1

1

STANDARDS 1,2,3


Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2)

A

B

D

C

Conjecture:

Emhh!..or



13

x

y

1

1

STANDARDS 1,2,3


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…?



14

STANDARDS 1,2,3


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



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!



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



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:



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


IF…, THEN …pq

CONVERSE:If they win competitions, then athletes train hard.

First convert to If…, then… statement

Now get the converse:



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




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



22

STANDARDS 1,2,3

INVERSE:The inverse of a conditional statement is when both the hypothesis and the conclusion are denied.

~p ~q



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:



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:



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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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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(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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(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.



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?



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.



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:



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:



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



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



36


ASSOCIATIVE PROPERTY:

Addition: (a + b) + c = a + (b + c)

(3 + 4) +1 = 3 + (4 + 1)

Multiplication:


34 45 6 = 34 45 6a b c= a b c

STANDARDS 1,2,3



37


IDENTITY PROPERTY:

Addition: a + 0 = 0 + a=a 5 + 0 = 0 + 5

Multiplication: 9 1 = 1 9


a 1 = 1 a = a

= 9

= 5

STANDARDS 1,2,3



38


INVERSE PROPERTY:

Addition: a + (-a) = (-a) + a=0 5 + (-5) = (-5) + 5

Multiplication:


= 1

= 1

= 0

a = a = 1 1a

1a

If a=0 then35

53

15

5

=53

35

15

= 5

STANDARDS 1,2,3



39


DISTRIBUTIVE PROPERTY:

Distributive:


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



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



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



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



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


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


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


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


49

STANDARDS 1,2,3

Given:

Prove:


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


50

STANDARDS 1,2,3

Given:

FBD is right

Prove:

ABF CBDand are complementary.


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


51

STANDARDS 1,2,3

Given:

Prove:


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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