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