Logic and Reasoning
Logic and Reasoning
Inductive Reasoning
Class Work
Make a conjecture about the missing term in the sequence based on the given numbers.
 12, 18, 24, 30, ?
 5, 10, 20, 40, 80,?
 24, 12, 6, 3, ?
 40, 30,20, 10, ?
 ?, 4, 12, 20, 28, 36
Make a conjecture based on the given statement.
 Segments are perpendicular.
 C is the midpoint of
 SQUA is a square
 TRI is a triangle
 A is four from B on a number line, B is at 3
 x=4
Homework
Make a conjecture about the missing term in the sequence based on the given numbers.
 8 , 24, ?, 216, 648
 30, 21, 12, 3, ?
 20,4,1, .25, ?
 25, ?, 5, 20, 35
 ?, 4, 12, 36, 108, 324
Make a conjecture based on the given statement.
 Segments are parallel
 C is the center of Circle C and P is on circle C
 RECT is a rectangle
 TRI is an equilateral triangle
 A is six from B on a number line, B is at 7
 xy=14 and x=7
Logic
Class Work
What is the validity of the following statements? If false, state a counterexample.
 The square root of a positive is a positive.
 Tomorrow is the start of a new month because today is the 30th.
 Gold weighs more than feathers.
Negate the following statement. What is the validity of the negation?
 2 + 2 = 6
 Albany is the capitol of New York.
 A square has 4 sides of the same length
What is the intersection of p and q? Draw a Venn diagram. State the members of p or q and
~p andq.
 p: multiples of 2 between 1 and 20; q: multiples of 3 between 1 and 20
 p: days of the week; q: usual days of the school year that there is no school
 p: squares; q: quadrilaterals
Create a truth table for each statement.
Homework
What is the validity of the following statements? If false, state a counterexample.
 The square root of a number is less than the number.
 Tomorrow is the end of the month because today is the 30th.
 Dallas is the capitol of Texas.
Negate the following statement. What is the validity of the negation?
 2 + 2 = 4
 Trenton is the capitol of New Jersey.
 A triangle has 3 sides of the same length
What is the intersection of p and q? Draw a Venn diagram. State the members of p or q
and ~p and q.
 p: multiples of 2 between 1 and 20; q: multiples of 4 between 1 and 20
 p: winter months; q: summer months
 p: even numbers; q: 1,2,3,4,5,6,7,8,9,10
Create a truth table for each statement.
IfThen Statements
Class Work
Identify the hypothesis with 1 line and the conclusion with 2 lines.
 If today is Tuesday, then tomorrow is Wednesday.
 If it rains today, then I’ll need an umbrella.
 If a quadrilateral is a square, then it has 4 equal sides.
 If x squared is 9, then x is three.
 XA=XB, if x is the midpoint of .
State the converse, inverse, and the contrapositive of the following conditional.
 If today is Tuesday, then tomorrow is Wednesday.
 If it rains today, then I’ll need an umbrella.
 If a quadrilateral is a square, then it has 4 equal sides.
 If x squared is 9, then x is three.
 XA=XB, if x is the midpoint of .
State the validity of the following conditional. State the converse, inverse, and the contrapositive of the following conditional and the state the validity of each.
 If a figure is a rectangle, then it has 4 sides.
Homework
Identify the hypothesis with 1 line and the conclusion with 2 lines.
 If I do my homework, then I can go to the movies.
 If I study my notes for an hour, then I’ll improve my test score.
 If triangle is isosceles, then it has at least 2 sides equal.
 If a number is squared, then the result is positive.
 4x+7=27, if x=5.
State the converse, inverse, and the contrapositive of the following conditional.
 If I do my homework, then I can go to the movies.
 If I do not skip homework problems, then I’ll improve my test score.
 If triangle is isosceles, then it has at least 2 sides equal.
 If a number is squared, then the product is positive.
 4x+7=27, if x=5.
State the validity of the following conditional. State the converse, inverse, and the contrapositive of the following conditional and the state the validity of each.
 If two angles are a linear pair, then the two angles are supplemental.
Deductive Reasoning
Class Work
Make a conclusion using the Law of Detachment.
 If someone runs a 4 minute mile at the track meet, then they will win.
Bob can run a 4 minute mile.
 If you plant geraniums, then you will have blooms all summer long.
I planted geraniums.
 If a quadrilateral is a square, then it will have 4 right angles
SQUA is a square.
Decide if the conclusion can be reached using the Law of Detachment.
 If today it rains, then tomorrow it will be sunny.
It is sunny today.
Conclusion: It rained yesterday.
 If you smile, then the whole world smiles with you.
John smiles.
Conclusion: The whole world smiles with John.
 If a natural number is multiplied by four, then the product is greater than the number.
½ is multiplied by 4.
Conclusion: The product of 4 and ½ is greater than 1/2.
Make a conclusion using the Law of Syllogism.
 If I work hard in math, then I’ll get a good grade.
If I get a good grade, then I will go to a good college.
 You will get 12 doughnuts, if you by a dozen.
If you have 12 doughnuts, then you can share them with friends.
 If 2 lines are perpendicular, then 4 right angles are formed.
If you have 4 right angles, then they are all congruent.
Decide if the conclusion can be reached using the Law of Syllogism.
 If a number is a natural number, then it is real.
If a number is an integer, then it is real.
Conclusion: Natural numbers are integers.
 If x = 4, then x2=16.
If x2 = 16, then (x2)2 = 256.
Conclusion: If x=4, then (x2)2 = 256.
 If zigs are zogs, then zogs are zags.
If zags are zegs, then zegs are zugs.
Conclusion: If zigs are zogs, then zegs are zugs.
Homework
Make a conclusion using the Law of Detachment.
 If Tim passes his final, then he will pass for the year.
Tim passed his final.
 If you remember to say please and thank you, then you are polite.
Peggy always remembers to say please and thank you.
 If a line passes through the center of a circle, then it contains a diameter.
contains P, the center of circle P.
Decide if the conclusion can be reached using the Law of Detachment.
 If today it rains, then the game will be cancelled.
The game was cancelled.
Conclusion: It rained today.
 If x=4, then x2=16.
x= 4.
Conclusion: x2=16.
 If a two rays share an endpoint, then they form an angle.
share an endpoint.
Conclusion: form an angle.
Make a conclusion using the Law of Syllogism.
 If today is Friday, then tomorrow is Saturday.
If tomorrow is Saturday, then yesterday was Thursday.
 You can buy lunch, if you have $5.
If you buy lunch, then you don’t have to bring lunch.
 If an angle is acute, then the angle is less than 90.
If you bisect a right angle, then you have an acute angle .
Decide if the conclusion can be reached using the Law of Syllogism.
 If a number is a natural number, then it is an integer.
If a number is an integer, then it is real.
Conclusion: If 3 is a natural number, then 3 is a real.
 If a triangle has 2 equal sides, then it isosceles.
If a triangle is isosceles, then it has 2 congruent angles.
Conclusion: If a triangle has 2 equal sides, then it has 2 congruent angles.
 If zigs are zogs, then zogs are zugs.
If zags are zigs, then zigs are zogs.
Conclusion: If zags are zigs, then zogs are zugs.
Intro to Proofs
Class Work
Prove the following by creating a tchart.
 Given: M is the midpoint of
Prove:
 Given:
Prove: B is the midpoint of
 Given: RECT is a rectangle
Prove:
 Write a paragraph proof for #93.
Homework
Prove the following by creating a tchart.
 Given: M is the midpoint of
Prove:
 Given:
Prove:
 Given: SQUA is a square
Prove: 
 Write a paragraph proof for #99.
Algebraic Proofs
Class Work
Given the first statement what reason justifies the second statement?
 1) 5x + 7 = 19Given
2) 5x = 12
 1) 4(x5)=20Given
2) 4x – 20=20
 1) Given
2) x=84
 1) a=b; 4a+5b=10Given
2) 4(b) +5b=10
 1) 11=xGiven
2) x=11
Prove the following by creating a tchart.
 Given: 3(x+11)=18 Prove: x= 5
 Given: =x+5Prove: x= 10.5
 Given: 10x 3(2x 4)=20Prove: x=2
 Write a paragraph proof for 106.
Homework
Given the first statement what reason justifies the second statement?
 1) 3x 8 = 20Given
2) 3x = 28
 1) 4(x5)=20Given
2) x5 = 5
 1) 8x+9=37Given
2) 8x=28
 1) 5a=b+9; b+9=10Given
2) 5a=10
 1) Given
2) 5(4x+2)=3(3x6)
Prove the following by creating a tchart.
 Given: 4x – 3(x5)=18 Prove: x= 3
 Given: =Prove: x= 4
 Given: 10x – 4(2x)+6=20Prove: x=7
 Write a paragraph proof for 116.
Proofs with Segments and Angles
Class Work
Find x and AN.
 N is between A and B. AN=2x, NB=4x, and AB= 24
 N is between A and B. AB = 5x9, BN= 3x+2, and NA= x2
 N is the midpoint AN= 2x+y, BN= x+3y, AB=20
Find x and
 lies on the interior of . =80, , and =3x.
 lies on the interior of . =7x20, , and =x+6.
 bisects . =52, , and =3x – 2y.
Prove the following by creating a tchart.
 Given: AB=XY and BC=YZ
Prove: AC=XZ
 Given: AB=CD
M is the midpoint of
N is the midpoint of
Prove:
 Given:
Prove:
 Given:
Prove:
 Write a paragraph proof for 128
Homework
Find x and AN.
 N is between A and B. AN=3x, NB=5x6, and AB= 26
 N is between A and B. AB = 5x, BN= 3x6, and NA= x+12
 N is the midpoint AN= 6x+2y, BN= 5x+6y, AB=104
Find x and
 lies on the interior of . =60, , and =4x.
 lies on the interior of . =3x10, , and =x2.
 bisects . =90, , and =5x – 2y.
Prove the following by creating a tchart.
 Given: AB=XY and AC=XZ
Prove: BC=YZ
 Given: AM=CN
M is the midpoint of
N is the midpoint of
Prove:
 Given:
Prove:
 Given:
Prove:
 Write a paragraph proof for 137
Multiple Choice
 Given: x2>0, Conjecture: x>0. Is the conjecture True of False, if false give a counter example.
a. Trueb. False, x= 2c. False, x=1/2d. False, x=0
 What is the inverse of: If x=3, then 2x=6
a. If 2x6, then x=3b. If 2x6, then x3 c. If x3, then 2x6 d. If 2x=6, then x=3
 If gold is pure then its 24 karat. Jen’s ring is pure gold. We can conclude that her ring is 24 karat. This is an example of
a. Contrapositiveb. Law of Syllogismc. Law of Detachment d. None of these
 What property justifies: If 3x2=8, then 3x=10.
a. Substitutionb. Additionc. Divisiond. Transitive
 What property justifies: If =8, then 4x5=48.
a. Distributionb. Additionc. Multiplicationd. Transitive
 D is between S and T. DS= 4x+8, ST=7x , and DT=2x3. Find DT
a. 5b. 7c. 28d. 35
 lies on the interior of , , , and
Find
a. .5b. 3c. 11.4d. 25
 What is the reason that allows statement 2 to be made?
Given
?
a. Definition of perpendicular
b. Definition of a right angle
c. Addition Property
d. Definition of Complementary
 What is the reason that allows statement 2 to be made?
Given
?
a. Addition Property
b. Definition of a Midpoint
c. Betweenness Theorem
d. Segment Bisector
 When is a true statement?
a. when p is true and q is true
b. when p is true or q is true
c. when p is false and q is true
d. when p is false or q is true
 The contrapositive of: If toady is Monday, then tomorrow is Tuesday, is
a. If today is Tuesday, then yesterday was Monday.
b. If tomorrow isn’t Tuesday, then yesterday wasn’t Monday.
c. If tomorrow isn’t Tuesday, then today isn’t Monday.
d. If today isn’t Monday, then tomorrow isn’t Tuesday.
 What is the hypotheses of
It’s going to be a great day, if you get up on the right side of the bed.
a. It’s going to be a great day
b. You get out on the right side of the bed
c. You got a good night sleep
d. Today is not a great day
Open Ended
 a. Create a truth table for:
b. What is the validity of p is true, q is false, and r is true?
c. How can a truth table be used to show two statements are equivalent?
 Create a twocolumn proof for the solution of
 Refer to the statement: All quadrilaterals are rectangles.
 Write a conditional based on this statement.
 Is the conditional in part a true or false, if false give a counter example.
 Write the contrapositive of the conditional from part a.
Answers
 Add 6; 36
 Multiply by 2; 160
 Div. by 2; 1.5
 Add 10; 0
 Add 8; 4
 Make a right angle
 Xc=cy
 All sides are equal
 Has 3 sides
 A is 1 or 7
 X2=16
 Mult 3; 72
 Subtract 9; 6
 Divide by 4; .0625
 Add 15: 10
 Mult 3; 4/3
 AB and CD have no points in common
 CP is a ridus
 RE ≥ CT
 TR=RI =TI
 B is 1 or 13
 Y=2
 True
 False, it could be October 30th
 False, it depends how many pounds of each
 2+2 ≠6 True
 Albany is not the capital of new York. False
 A Square does not have four sides of the same length. False
 P^q: {6,12,18} ; p U q {2,3,4,6,8,9,10,12,14,15,16,18,20} ; p ^q {3,9,15}
[Type a quote from the document or the summary of an interesting point. You can position the text box anywhere in the document. Use the Text Box Tools tab to change the formatting of the pull quote text box.]
(middle= 6, 12,8 for diagram above)
 P^q { SA SU}; p u q { SA SU MO TU WE TH FR}; p ^q { mon tues wed thur fri}
 P ^q { 14}; p u q{2,5,6,8,.10,11,14,17,18,20}; _ p ^q { 5,8,11,17,20}
Middle= 2, 14
P / Q / P u qT / T / T
T / F / T
F / T / T
F / F / F
P / P / Q / P^q
T / F / T / F
T / F / F / F
F / T / T / T
F / T / F / F
P / Q / R / Q^R / P^ / Q^R
T / T / T / T / T
T / T / F / F / F
T / F / T / F / F
T / F / F / F / F
F / T / T / T / F
F / T / F / F / F
F / F / T / F / F
F / F / F / F / f
 False
 False, today could be sept 30th
 False, austin is the capital
 2+2 ≠4 false
 Trenton is not the capital of NJ, false
 A triangle does not have 3 sides the same length; false
 P^q {4,8,12,16,20}; p u q {2,4,6,8,18,20}; {2,6,10,14,18)
 P ^q: none; p u q: {dec,jan, feb, jue, july aug}; ~p ^ q; {June july aug}
PQ
 P^q: {2,4,8,10}; p u q: {1,2,3,4,5,6,7,8,9,10,12,14,16,18,20} ~p ^ q {1,3,5,7,9}
Middle= 2, 4,6, 8 10
P / Q / P^qT / T / T
T / F / F
F / T / F
F / F / F
P / Q / ~P / ~Q / ~P U q
T / T / F / F / F
T / F / F / T / T
F / T / T / F / T
F / F / T / T / T
P / Q / R / ~Q / ~R / ~Q u ~R / P ^( ~QU ~R)
T / T / T / F / F / F / F
T / T / F / F / T / T / T
T / F / T / T / F / T / T
T / F / F / T / T / T / T
F / T / T / F / F / F / F
F / T / F / F / T / T / F
F / F / T / T / F / T / F
F / F / F / T / T / T / F
 If today is Tuesday, then tomorrow is Wednesday.
 If it rains today, then I’ll need an umbrella.
 If a quadrilaterial is a square, then it has four equal sides
 If x squared is 9, then x is three.
 XA=XB, if x is the midpoint of AB.
 Conv: If tomorrow is wed, then today is tues; inv: If today is not tues, then tomorrow isn’t wed; Cont: If tomorrow isn’t wed, then today isn’t tues.
 Conv: If I’ll need an umbrella, then it rains today; Inv: If it doesn’t rain today, then I won’t need an umbrella; Cont: If I don’t need an umbrella, then it didn’t rain today
 Con: If it has four sides, then a quadrilateral is a square; Inv: If a quad is not a square, then it doesn’t have four sides; Cont: If a quad has four equal sides, then it is not a square
 Con: If x=3, then x2=9; Inv: if x2=9, then x≠3; Cont: if x≠3, then x2≠9
 Con: If XA=XB, then x is the midpt of AB; Inv: If x is not the midpt of AB, then XA ≠XB; Cont: If XA ≠XB, then x is ot the midpt of AB
 Statement is true; Conv: If a figure has four sides, then it is a rectangle; fls; INV: If a figure is not a rect, then it doesn’t have four sides; false; Cont: If a figure doesn’t have four sides, then it is not a rect; true
 If I do my homework, then I can go to the movies.
 If I study my notes for an hour, then I’ll improve my test score.
 If a triangle is isosceles, then it has at least two sides equal.
 If a number is squared, then the result is positive.
 4x+7=27, if x=5.
 CON: if I can go to the movies, then I do my homework; INV: if I don’t do my homework, then I cant go to the movies; CONTR: if I cant go to the movies, then I didn’t do my homework.
 CON: if I improve test scores, then I didn’t skip homework problems; INV: if I skip homework problems, then I wont improve my test scores; CONTR: if I don’t improve test scores, then I skipped homework problems.
 CON: if a triangle has at least 2 sides equal, then it is isosceles; INV: if a triangle is not isosceles, then it doesn’t have at least two sides.; CONTR: If a triangle does not have at least two sides equal, then it is not isosceles.
 CON: if the product is positive, then a number is squared; INV: if a number isn’t squared, then the product is not positive. CONTR: if the product isn’t positive, then a number isn’t squared.
 Conv: if 4x+7=27, then x=5’ Inv: If x≠5, then 4x+7 ≠27. Contr: If 4x+7≠27, then x≠5
 Statement is true; conv: if two angles are supplemental, then the angles are a linear pair (false); Inv: if two angles are not a linear pair, then they are not supplemental (false); Contr: If two angles are not supplemental, then they are not a linear pair (true)
 Bob will win
 I will have blooms all summer long
 Square has four right angles
 No: p→q +q is true
 Yes
 No
 If I work hard in math then I will go to a good college.
 Not possible
 If two lines are perpendicular, then four angles are congruent.
 No
 Yes
 No
 Tim will pass for the year
 Peggy is polite
 AB contains a diameter
 No
 No
 Yes
 If today is Friday then yesterday was Thursday
 If you have five dollars, then you don’t have to bring lunch.
 If you bisect an angle then the angle is less than 90 degrees.
 Yes
 Yes
 Yes
Statement / Reason
N is midpoint of AU / Given
AMMU / Def of midpt
AM MN / Def of congruence
Statement / Reason
XB BY / Given
XB = BY / Def of congruence
B is midpt of XY / Def of midpt
Statement / Reason
RECT is a rectangle / Given
RE CT / Prop of rectangle
 Dt is given. N is the midpoint of AV. AM MV by definition of a midpoint. Using the definition of congruent AM = MV.
Statement / Reason
M is midpoint o AN / Given
AM = AN / Def of midpoint
Statement / Reason
XB BY BY YC / Given
XB YC / Transitive property
Statement / Reason
SQUA is a square / Given
<Q is a right angle / Prop of a square
SQ is perpendicular to QU / Def of perpendicular
 SQUA is given to be a square. A property of a square is that it has four right angles, therefore <Q is a right angle. SQ is perpendicular to QU by the definition of perpendicular.
 Addition (subtraction) property of equality
 Distribution
 Multiplication property of equality
 substitution
 Symmetry
Statement / Reason
3(x+11) =18 / Given
3x+33=18 / Distribution
3x=15 / Add prop of eq.
X=5 / Division prop of eq.
Statement / Reason
=x+5 / Given
4x+9=6x+30 / Mult prop of eq.
21=2x / Add prop of eq.
10.5 =x / Mult (division) prop
X=10.5 / symmetric
Statement / Reason
10x3(2x4)=20 / Given
10x6x+12=20 / Distribution
4x+12=20 / Addition
4x=8 / Addition (subtraction) prop of eq
X=2 / Multiplication (division) prop of eq
 Given 3(x+11)=18, use distribution property to get 3x+33=18. 3x=15, results from using addition property of equality. The solution of x=5 is found by applying multiplication property.
 Addition property
 Multiplication property
 Addition property
 Transitive property
 Multiplication property
Statement / Reason
4x+3(x5)=18 / Given
4x3x+15=18 / Distribution
X+15=18 / Addition
X=3 / Addition prop of eq
Statement / Reason
/ Given
3(2x+6)=2(2x1) / Mult prop of eq
9x+18 =4x2 / Distribution prop
5x=20 / Addition prop of eq
X=40 / multiplication
Statement / Reason
10x4(2x)+6=20 / Given
10x8x+6=20 / Multiplication
2x+6=20 / Addition
2x=14 / Addition prop of eq
X=7 / Multiplication prop

is given. Cross multiply to get 3(3x+6) =2(2x7). Distribution leads to 9x+18=4x+2. 5x=20 is arrived at by using addition prop of eq to add 4x to both sides and 18 to both sides. Using multiplication property of equality, x=4.  X=4, AN =8
 X=9, AN=7
 X=4 AN =10
 X=16, m<ABX=32
 X=7, m<ABX =23
 X=10, m<ABX =26
Statement / Reason
AB=x4 and BC=42 / Given
AB+BC =x4+47 / Addition prop of eq
AB+BC=AC / Addition
X4+47=x7 / ADD
AC=x7 / substitution
Statement / Reason
AB=CD
M is midpt of AB
N is midpt of CD / Given
AM=MB, CN=ND / Def of midpt
AM+MB =AB, CN+ND=CD / Segment addition
AM+MB = CN+ND / Substitution
AM+AM =CN +CN / Substation
2AM=@CN / Addition
AM=CN / Mult prop of eq
Statement / Reason
<ABC <XYZ / Given
M<ABC + m<EBC =m<ABC / Angle addition
M<ABC = M<EBC = m<XYF + m<FYZ / Substitution
BE bisects <ABC
4f bisects <XYZ / Given
M<ABC = m<EBC M<XYZ = m<FYZ / Def of bisects
M<ABC =m<ABE = m<XYZ +m<XYZ / Substit five into 3
2m<ABE = 2m<XYZ / Addition
M<ABE = m<XYF / Mult prop
<ABE / Def of congruent
Statement / Reason
<ABF / Given
M<ABF = m<MNG
M<FBC= m<GNP / Def og congruent
M<ABF + m<FBC =m<MNG + m<GNP / Add prop
,<ABF = m<FBC = m<ABC / Angle add
M<ABC = m<MNP / Substitution
<ABC / Def of congruent
 X=4 AN=12
 X=6, AN=18
 X=8 AN=52
 X=11 m<ABX =16
 X=20 m<ABX =32
 X=10 m<ABS =45
Statement / Reason
AB=XY; AC=YZ / Given
AC=AB +BC
XZ=XY+YZ / Seg addition
AB+BC =XY +YZ / Substitution
BC=YZ / Add property of eq
 ]
Statement / Reason
M IS MIDPT OF AB
N IS MIDPT OF CD / GIVEN
AB = AM +MB
CD=CN+ND / SEG ADDITION
AM=MB
CN=ND / DEF OF MIDPT
AM=CN / GIVEN
2M=2CN / MULT PROP
AM+AM=CN+CN / ADD PROP
AM+MB = CN+ND / SUBST
AB=CD / SUBST
STATEMENT / REASON
BE BISECTS <ABC
YF BISECTS <XYZ / GIVEN
M<ABE = M<EBC
M<XYF = M<FYZ / DEF OF BISECTS
M<ABE = M<EBC / GIVEN
2M<ABE = 2M<EBC
MULTIPLE CHOICE
 B
 C
 C
 B
 C
 B
 D
 B
 C
 D
 C
 B
OPEN ENDED
 A)
p / Q / R / ~p / Q^r / ~p U (q^r)
T / T / T / F / T / T
T / T / F / F / F / F
T / F / T / F / F / F
T / F / F / F / F / F
F / T / T / T / T / T
F / T / F / T / F / T
F / F / T / T / F / T
F / F / F / T / F / T
b) false
C) If true and false for same inputs
Statements / Reason
2 (=11 / Given
=11 / Distribution
5x3=22 / Mult prop of eq
5x=25 / Add prop of eq
X=5 / Mult prop of eq
 A) if a figure is a quadrilateral, then it is a rectangle.
B) False, a trapezoid
C) if a figure is not a rectangle, then it is not a quadrilateral