Logic and Reasoning Inductive Reasoning

Class Work

Make a conjecture about the missing term in the sequence based on the given numbers.

1. 12, 18, 24, 30, ?

2. 5, 10, 20, 40, 80,?

3. 24, -12, 6, -3, ?

4. -40, -30,-20, -10, ?

5. ?, 4, 12, 20, 28, 36

Make a conjecture based on the given statement.

6. Segments AB̅̅ ̅̅ and AC̅̅̅̅ are perpendicular.

7. C is the midpoint of 𝑋𝑌̅̅ ̅̅

8. SQUA is a square

9. TRI is a triangle

10. A is four from B on a number line, B is at 3

11. x=4

Homework

Make a conjecture about the missing term in the sequence based on the given numbers.

12. 8 , 24, ?, 216, 648

13. 30, 21, 12, 3, ?

14. 20,-4,1, -.25, ?

15. -25, ?, 5, 20, 35

16. ?, 4, 12, 36, 108, 324

Make a conjecture based on the given statement.

17. Segments AB ⃡ and CD ⃡ are parallel

18. C is the center of Circle C and P is on circle C

19. RECT is a rectangle

20. TRI is an equilateral triangle

21. A is six from B on a number line, B is at 7

22. xy=14 and x=-7

Logic

Class Work

What is the validity of the following statements? If false, state a counterexample.

23. The square root of a positive is a positive.

24. Tomorrow is the start of a new month because today is the 30th.

25. Gold weighs more than feathers.

Negate the following statement. What is the validity of the negation?

26. 2 + 2 = 6

27. Albany is the capitol of New York.

28. 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.

29. p: multiples of 2 between 1 and 20; q: multiples of 3 between 1 and 20

30. p: days of the week; q: usual days of the school year that there is no school

31. p: squares; q: quadrilaterals

Create a truth table for each statement.

32. 𝑝 ∪ 𝑞

33. ~𝑝 ∩ 𝑞

34. 𝑝 ∪ (𝑞 ∩ 𝑟)

Homework

What is the validity of the following statements? If false, state a counterexample.

35. The square root of a number is less than the number.

36. Tomorrow is the end of the month because today is the 30th.

37. Dallas is the capitol of Texas.

Negate the following statement. What is the validity of the negation?

38. 2 + 2 = 4

39. Trenton is the capitol of New Jersey.

40. 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.

41. p: multiples of 2 between 1 and 20; q: multiples of 4 between 1 and 20

42. p: winter months; q: summer months

43. p: even numbers; q: 1,2,3,4,5,6,7,8,9,10

Create a truth table for each statement.

44. 𝑝 ∩ 𝑞

45. ~𝑝 ∪ ~𝑞

46. 𝑝 ∩ (~𝑞 ∪ ~𝑟)

If-Then Statements

Class Work

Identify the hypothesis with 1 line and the conclusion with 2 lines.

47. If today is Tuesday, then tomorrow is Wednesday.

48. If it rains today, then I’ll need an umbrella.

49. If a quadrilateral is a square, then it has 4 equal sides.

50. If x squared is 9, then x is three.

51. XA=XB, if x is the midpoint of AB̅̅ ̅̅ .

State the converse, inverse, and the contrapositive of the following conditional.

52. If today is Tuesday, then tomorrow is Wednesday.

53. If it rains today, then I’ll need an umbrella.

54. If a quadrilateral is a square, then it has 4 equal sides.

55. If x squared is 9, then x is three.

56. XA=XB, if x is the midpoint of AB̅̅ ̅̅ .

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.

57. If a figure is a rectangle, then it has 4 sides.

Homework

Identify the hypothesis with 1 line and the conclusion with 2 lines.

58. If I do my homework, then I can go to the movies.

59. If I study my notes for an hour, then I’ll improve my test score.

60. If triangle is isosceles, then it has at least 2 sides equal.

61. If a number is squared, then the result is positive.

62. 4x+7=27, if x=5.

State the converse, inverse, and the contrapositive of the following conditional.

63. If I do my homework, then I can go to the movies.

64. If I do not skip homework problems, then I’ll improve my test score.

65. If triangle is isosceles, then it has at least 2 sides equal.

66. If a number is squared, then the product is positive.

67. 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.

68. 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.

69. If someone runs a 4 minute mile at the track meet, then they will win.

Bob can run a 4 minute mile.

70. If you plant geraniums, then you will have blooms all summer long.

I planted geraniums.

71. 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.

72. If today it rains, then tomorrow it will be sunny.

It is sunny today.

Conclusion: It rained yesterday.

73. If you smile, then the whole world smiles with you.

John smiles.

Conclusion: The whole world smiles with John.

74. 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.

75. 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.

76. You will get 12 doughnuts, if you by a dozen.

If you have 12 doughnuts, then you can share them with friends.

77. 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.

78. 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.

79. If x = 4, then x2=16.

If x2 = 16, then (x2)2 = 256.

Conclusion: If x=4, then (x2)2 = 256.

80. 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.

81. If Tim passes his final, then he will pass for the year.

Tim passed his final.

82. If you remember to say please and thank you, then you are polite.

Peggy always remembers to say please and thank you.

83. If a line passes through the center of a circle, then it contains a diameter.

𝐴𝐵 ⃡ contains P, the center of circle P.

Decide if the conclusion can be reached using the Law of Detachment.

84. If today it rains, then the game will be cancelled.

The game was cancelled.

Conclusion: It rained today.

85. If x=4, then x2=16.

x= -4.

Conclusion: x2=16.

86. If a two rays share an endpoint, then they form an angle.

𝐸𝐹 𝑎𝑛𝑑 𝐸𝐺 share an endpoint.

Conclusion: 𝐸𝐹 𝑎𝑛𝑑 𝐸𝐺 form an angle.

Make a conclusion using the Law of Syllogism.

87. If today is Friday, then tomorrow is Saturday.

If tomorrow is Saturday, then yesterday was Thursday.

88. You can buy lunch, if you have $5.

If you buy lunch, then you don’t have to bring lunch.

89. 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.

90. 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.

91. 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.

92. 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 t-chart.

93. Given: M is the midpoint of 𝐴𝑉̅̅ ̅̅

Prove: 𝐴𝑀̅̅̅̅̅ ≅ 𝑀𝑉̅̅̅̅̅

94. Given: 𝑋𝐵̅̅ ̅̅ ≅ 𝐵𝑌̅̅ ̅̅

Prove: B is the midpoint of XY̅̅̅̅

95. Given: RECT is a rectangle

Prove: 𝑅𝐸̅̅ ̅̅ ≅ 𝐶𝑇̅̅̅̅

96. Write a paragraph proof for #93.

Homework

Prove the following by creating a t-chart.

97. Given: M is the midpoint of 𝐴𝑁̅̅ ̅̅

Prove: 𝐴𝑀 = 𝐴𝑁

98. Given: 𝑋𝐵̅̅ ̅̅ ≅ 𝐵𝑌̅̅ ̅̅ ; 𝐵𝑌̅̅ ̅̅ ≅ 𝑌𝐶̅̅̅̅

Prove: 𝑋𝐵̅̅ ̅̅ ≅ 𝑌𝐶̅̅̅̅

99. Given: SQUA is a square

Prove: 𝑆𝑄̅̅̅̅ | 𝑄𝑈̅̅ ̅̅ ̅

100. Write a paragraph proof for #99.

Algebraic Proofs

Class Work

Given the first statement what reason justifies the second statement?

101. 1) 5x + 7 = 19 Given

2) 5x = 12

102. 1) 4(x-5)=20 Given

2) 4x – 20=20

103. 1) 𝑥

4= 21 Given

2) x=84

104. 1) a=b; 4a+5b=10 Given

2) 4(b) +5b=10

105. 1) 11=x Given

2) x=11

Prove the following by creating a t-chart.

106. Given: 3(x+11)=18 Prove: x= -5

107. Given: 4x+9

6=x+5 Prove: x= -10.5

108. Given: 10x -3(2x -4)=20 Prove: x=2

109. Write a paragraph proof for 106.

Homework

Given the first statement what reason justifies the second statement?

110. 1) 3x -8 = 20 Given

2) 3x = 28

111. 1) 4(x-5)=20 Given

2) x-5 = 5

112. 1) 8x+9=37 Given

2) 8x=28

113. 1) 5a=b+9; b+9=10 Given

2) 5a=10

114. 1) 4𝑥+2

3=

3𝑥−6

5 Given

2) 5(4x+2)=3(3x-6)

Prove the following by creating a t-chart.

115. Given: 4x – 3(x-5)=18 Prove: x= 3

116. Given: 3x+6

2=

2x−1

3 Prove: x= -4

117. Given: 10x – 4(2x)+6=20 Prove: x=7

118. Write a paragraph proof for 116.

Proofs with Segments and Angles

Class Work

Find x and AN.

119. N is between A and B. AN=2x, NB=4x, and AB= 24

120. N is between A and B. AB = 5x-9, BN= 3x+2, and NA= x-2

121. N is the midpoint AN= 2x+y, BN= x+3y, AB=20

Find x and 𝑚∠𝐴𝐵𝑋.

122. 𝐵𝑋 lies on the interior of ∠𝐴𝐵𝐶. 𝑚∠𝐴𝐵𝐶=80, 𝑚∠𝐴𝐵𝑋 = 2𝑥, and

𝑚∠𝐶𝐵𝑋=3x.

123. 𝐵𝑋 lies on the interior of ∠𝐴𝐵𝐶. 𝑚∠𝐴𝐵𝐶=7x-20, 𝑚∠𝐴𝐵𝑋 = 3𝑥 + 2, and

𝑚∠𝐶𝐵𝑋=x+6.

124. 𝐵𝑋 bisects ∠𝐴𝐵𝐶. 𝑚∠𝐴𝐵𝐶=52, 𝑚∠𝐴𝐵𝑋 = 2𝑥 + 3𝑦, and 𝑚∠𝐶𝐵𝑋=3x – 2y.

Prove the following by creating a t-chart.

125. Given: AB=XY and BC=YZ

Prove: AC=XZ

126. Given: AB=CD

M is the midpoint of AB̅̅ ̅̅

N is the midpoint of CD̅̅̅̅

Prove: AM̅̅̅̅̅ ≅ CN̅̅ ̅̅

127. Given: ∠𝐴𝐵𝐶 ≅ ∠𝑋𝑌𝑍; 𝐵𝐸 𝑏𝑖𝑠𝑒𝑐𝑡𝑠∠𝐴𝐵𝐶; 𝑌𝐹 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑋𝑌𝑍

Prove: ∠𝐴𝐵𝐸 ≅ ∠𝑋𝑌𝐹

128. Given: ∠𝐴𝐵𝐹 ≅ ∠𝑀𝑁𝐺; ∠𝐹𝐵𝐶 ≅ ∠𝐺𝑁𝑃

Prove: ∠𝐴𝐵𝐶 ≅ ∠𝑀𝑁𝑃

129. Write a paragraph proof for 128

Homework

Find x and AN.

130. N is between A and B. AN=3x, NB=5x-6, and AB= 26

131. N is between A and B. AB = 5x, BN= 3x-6, and NA= x+12

132. N is the midpoint AN= 6x+2y, BN= 5x+6y, AB=104

Find x and 𝑚∠𝐴𝐵𝑋.

133. 𝐵𝑋 lies on the interior of ∠𝐴𝐵𝐶. 𝑚∠𝐴𝐵𝐶=60, 𝑚∠𝐴𝐵𝑋 = 2𝑥 − 6, and

𝑚∠𝐶𝐵𝑋=4x.

134. 𝐵𝑋 lies on the interior of ∠𝐴𝐵𝐶. 𝑚∠𝐴𝐵𝐶=3x-10, 𝑚∠𝐴𝐵𝑋 = 𝑥 + 12, and

𝑚∠𝐶𝐵𝑋=x-2.

135. 𝐵𝑋 bisects ∠𝐴𝐵𝐶. 𝑚∠𝐴𝐵𝐶=90, 𝑚∠𝐴𝐵𝑋 = 4𝑥 + 2𝑦, and 𝑚∠𝐶𝐵𝑋=5x – 2y.

Prove the following by creating a t-chart.

136. Given: AB=XY and AC=XZ

Prove: BC=YZ

137. Given: AM=CN

M is the midpoint of AB̅̅ ̅̅

N is the midpoint of CD̅̅̅̅

Prove: MB̅̅ ̅̅ ≅ ND̅̅ ̅̅

138. Given: ∠𝐴𝐵𝐸 ≅ ∠𝑋𝑌𝐹; 𝐵𝐸 𝑏𝑖𝑠𝑒𝑐𝑡𝑠∠𝐴𝐵𝐶; 𝑌𝐹 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑋𝑌𝑍

Prove: ∠𝐴𝐵𝐶 ≅ ∠𝑋𝑌𝑍

139. Given: ∠𝐴𝐵𝐹 ≅ ∠𝑀𝑁𝐺; ∠𝐹𝐵𝐶 ≅ ∠𝐺𝑁𝑃

Prove: ∠𝐴𝐵𝐶 ≅ ∠𝑀𝑁𝑃

140. Write a paragraph proof for 137

Multiple Choice 1. Given: x2>0, Conjecture: x>0. Is the conjecture True of False, if false give a counter

example.

a. True b. False, x= -2 c. False, x=1/2 d. False, x=0

2. What is the inverse of: If x=3, then 2x=6

a. If 2x≠6, then x=3 b. If 2x≠6, then x≠3 c. If x≠3, then 2x≠6 d. If 2x=6, then x=3

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. Contrapositive b. Law of Syllogism c. Law of Detachment d. None of these

4. What property justifies: If 3x-2=8, then 3x=10.

a. Substitution b. Addition c. Division d. Transitive

5. What property justifies: If 4x−5

6=8, then 4x-5=48.

a. Distribution b. Addition c. Multiplication d. Transitive

6. D is between S and T. DS= 4x+8, ST=7x , and DT=2x-3. Find DT

a. 5 b. 7 c. 28 d. 35

7. 𝐴𝑋 lies on the interior of ∠𝑀𝐴𝐷, 𝑚∠𝑀𝐴𝑋 = 22, 𝑚∠𝑋𝐴𝐷 = 5𝑥 + 10, and

𝑚∠𝑀𝐴𝐷 = 15𝑥 + 2. Find 𝑚∠𝑋𝐴𝐷

a. .5 b. 3 c. 11.4 d. 25

8. What is the reason that allows statement 2 to be made?

∠𝐴𝐵𝐶 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 Given

𝑚∠𝐴𝐵𝐶 = 90 ?

a. Definition of perpendicular

b. Definition of a right angle

c. Addition Property

d. Definition of Complementary

9. What is the reason that allows statement 2 to be made?

𝐵 𝑖𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐴 𝑎𝑛𝑑 𝐶 Given

𝐴𝐵̅̅ ̅̅ + 𝐵𝐶̅̅ ̅̅ = 𝐴𝐶̅̅ ̅̅ ?

a. Addition Property

b. Definition of a Midpoint

c. Betweenness Theorem

d. Segment Bisector

10. When is ~𝑝 ∪ 𝑞 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

11. 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.

12. 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 1. a. Create a truth table for: ~𝑝 ∪ (𝑞 ∩ 𝑟)

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?

2. Create a two-column proof for the solution of 2 (5x−3

4) = 11

3. Refer to the statement: All quadrilaterals are rectangles.

a. Write a conditional based on this statement.

b. Is the conditional in part a true or false, if false give a counter example.

c. Write the contrapositive of the conditional from part a.

Answers 1. Add 6; 36

2. Multiply by 2; 160

3. Div. by -2; 1.5

4. Add 10; 0

5. Add 8; -4

6. Make a right angle

7. Xc=cy

8. All sides are equal

9. Has 3 sides

10. A is -1 or 7

11. X2=16

12. Mult 3; 72

13. Subtract 9; -6

14. Divide by -4; .0625

15. Add 15: -10

16. Mult 3; 4/3

17. AB and CD have no points in common

18. CP is a ridus

19. RE ≥ CT

20. TR=RI =TI

21. B is 1 or 13

22. Y=-2

23. True

24. False, it could be October 30th

25. False, it depends how many pounds of

each

26. 2+2 ≠6 True

27. Albany is not the capital of new York.

False

28. A Square does not have four sides of

the same length. False

29. 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)

30. P^q { SA SU}; p u q { SA SU MO TU WE

TH FR}; -p ^q { mon tues wed thur fri}

31. 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 q

T T T

T F T

2 10 20 4 14 8

16 3 9 15

2 6 10 18

5 8 11 17 20

S A SU

Mon, Tues,

Wed, Thurs, Fri

F T T

F F F

32.

33.

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

34.

35. False √1

4= 1/2

36. False, today could be sept 30th

37. False, austin is the capital

38. 2+2 ≠4 false

39. Trenton is not the capital of NJ, false

40. A triangle does not have 3 sides the

same length; false

41. P^q {4,8,12,16,20}; p u q

{2,4,6,8,18,20}; {2,6,10,14,18)

42. P ^q: none; p u q: {dec,jan, feb, jue, july

aug}; ~p ^ q; {June july aug}

P Q

43. 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^q

T T T

T F F

F T F

F F F

44.

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

45.

P Q R ~Q ~R ~Q u ~R

P ^( ~QU ~R)

T T T F F F F

12 14 16 18

20

1 3 5 7 9

4 8 12 16

20

P q 2 6 10 14 18

Dec, Jan, Feb

June, July, Aug

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

46.

47. If today is Tuesday, then tomorrow is

Wednesday.

48. If it rains today, then I’ll need an

umbrella.

49. If a quadrilaterial is a square, then it has

four equal sides

50. If x squared is 9, then x is three.

51. XA=XB, if x is the midpoint of AB.

52. 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.

53. 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

54. 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

55. Con: If x=3, then x2=9; Inv: if x2=9, then

x≠3; Cont: if x≠3, then x2≠9

56. 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

57. 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

58. If I do my homework, then I can go to

the movies.

59. If I study my notes for an hour, then I’ll

improve my test score.

60. If a triangle is isosceles, then it has at

least two sides equal.

61. If a number is squared, then the result

is positive.

62. 4x+7=27, if x=5.

63. 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.

64. 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.

65. 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.

66. 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.

67. Conv: if 4x+7=27, then x=5’ Inv: If x≠5,

then 4x+7 ≠27. Contr: If 4x+7≠27, then

x≠5

68. 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)

69. Bob will win

70. I will have blooms all summer long

71. Square has four right angles

72. No: p→q +q is true

73. Yes

74. No

75. If I work hard in math then I will go to a

good college.

76. Not possible

77. If two lines are perpendicular, then four

angles are congruent.

78. No

79. Yes

80. No

81. Tim will pass for the year

82. Peggy is polite

83. AB contains a diameter

84. No

85. No

86. Yes

87. If today is Friday then yesterday was

Thursday

88. If you have five dollars, then you don’t

have to bring lunch.

89. If you bisect an angle then the angle is

less than 90 degrees.

90. Yes

91. Yes

92. Yes

93.

Statement Reason

N is midpoint of AU

Given

AM≅MU Def of midpt

AM ≅MN Def of congruence

94.

Statement Reason

XB ≅BY Given

XB = BY Def of congruence

B is midpt of XY Def of midpt

95.

Statement Reason

RECT is a rectangle Given

RE ≅CT Prop of rectangle

96. 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

97.

Statement Reason

XB ≅BY BY ≅YC Given

XB ≅ YC Transitive property

98.

Statement Reason

SQUA is a square Given

<Q is a right angle Prop of a square

SQ is perpendicular to QU

Def of perpendicular

99.

100. 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.

101. Addition (subtraction) property

of equality

102. Distribution

103. Multiplication property of

equality

104. substitution

105. Symmetry

106.

Statement Reason

3(x+11) =18 Given

3x+33=18 Distribution

3x=-15 Add prop of eq.

X=-5 Division prop of eq.

107.

Statement Reason 4𝑥+9

6 =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

108.

Statement Reason

10x-3(2x-4)=20 Given

10x-6x+12=20 Distribution

4x+12=20 Addition

4x=8 Addition (subtraction) prop of eq

X=2 Multiplication (division) prop of eq

109. 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.

110. Addition property

111. Multiplication property

112. Addition property

113. Transitive property

114. Multiplication property

115.

Statement Reason

4x+3(x-5)=18 Given

4x-3x+15=18 Distribution

X+15=18 Addition

X=-3 Addition prop of eq

116.

Statement Reason 3𝑥 + 6

2=

2𝑥 − 1

3

Given

3(2x+6)=2(2x-1) Mult prop of eq

9x+18 =4x-2 Distribution prop

5x=-20 Addition prop of eq

X=-40 multiplication

117.

Statement Reason

10x-4(2x)+6=20 Given

10x-8x+6=20 Multiplication

2x+6=20 Addition

2x=14 Addition prop of eq

X=7 Multiplication prop

118. 3𝑥+6

2=

2𝑥−1

3 is given. Cross multiply to

get 3(3x+6) =2(2x-7). 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.

119. X=4, AN =8

120. X=9, AN=7

121. X=4 AN =10

122. X=16, m<ABX=32

123. X=7, m<ABX =23

124. 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

125.

126.

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 ≅ < 𝑋𝑌𝐹 Def of congruent

127.

Statement Reason

<ABF ≅ <𝑀𝑁𝐺; < 𝐹𝐵𝐶 ≅ < 𝐺𝑁𝑃

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 ≅ < 𝑀𝑁𝑃 Def of congruent

128.

129.

130. X=4 AN=12

131. X=6, AN=18

132. X=8 AN=52

133. X=11 m<ABX =16

134. X=20 m<ABX =32

135. 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

136. ]

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

137.

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

138.

139.

140.

MULTIPLE CHOICE

1. B

2. C

3. C

4. B

5. C

6. B

7. D

8. B

9. C

10. D

11. C

12. B

OPEN ENDED

1. 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

2.

Statements Reason

2 (5𝑥−3

4=11 Given

5𝑥−3

2 =11 Distribution

5x-3=22 Mult prop of eq

5x=25 Add prop of eq

X=5 Mult prop of eq

3. 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