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Review Chapter 2: Logic Reasoning
Review Distance Formula.
1. Given the points find the distance. Round to the nearest hundredths.
a. (3,-4) and (-2, 7) ~
b. (- 11,0) and (3,0)-
Sec 2-1 Using Inductive Reasoning to Make Conjectures
2. A statement that you believe to be true based on inductive reasoning is called a(n)
fom
3. Find the next item in the pattern: 0, 1, 1,2, 3, 5, 8, 13,...
' ef~Write the rule:
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4. Find the next item in the pattern: /
Write the rule: -fk-£ t-^vi^LUJL
4€rry>s ft <$<-+-
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\
I A i ^ 6A//frvO-/r"yi
Complete each conjecture.
5. The product of an even number and an odd number is
L ^'C. n
(Give at least 3 examples to justify your answer).
6. The number of lines formed by four points, no three of, which are noncolliner, is
(Draw a picture to justify your answer).
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7. Determine if the conjecture is true or false. If not, write or draw a COUNTEREXAMPLE
If C is the midpoint of AB , then AC = BC ._
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8. Which of the following conjectures is false?
Write an example next to each conjecture to support each answer choice.
a) If x is odd, then x + 1 is even.
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b) The sum of two odd numbers is even.
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c) The difference of two even numbers is positive.
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d) If x is positive, then -x is negative.
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Sec 2-2 Conditional Statements
9. Underline thejiypothjesis,_anjd_.ciic]e the conclusion of the statement:
"Sue will watch the Rose Bowl if today is January 1st. "
10. Identify the hypothesis and conclusion of the conditional statement:
"Two angles whose sum is 180°<are supplementary."
Hypothesis: "K^O QnfrllS HhOS<L Sory> iX
Conclusion:
> Many sentences without the words if and then can be written as conditionals. To do so, identify
the sentence's hypothesis and conclusion by figuring out which part of the statement depends on
the other.
11. Write a conditional statement from the sentence "An angle that measures 90°
angle".
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12. Determine if the conditional statement is true. If false, give a counterexample.
"If6x-2 = 4x + 12, thenx = 3"
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13. Write the converse, inverse, and contrapositive of the statement.
(Find the truth value of each statement)
"If two angles are adjacent, then they have a common ray and non common interior points".
Converse: / /- ~h^O gnqj&y h&vC- (\ C&mmun r^^\ and ndn common in Inverse: l£ -fWD an/-,] \f^ <3re nOl adjaf0n-l-fi-kttj.
^On'4 ^^(Vf' a Common hzu^ ov
v,p -^7<\x^ no^-T CorlomDn i*^ 4d^"c>r yjp t~/4Contrapositive: // fv-jo /?/?^//.S /7^?/7 ^ hGvp a ( OfYlmriin rza,j ^ (^otnsyx}^ t ^•
> Related conditional statements that have the same truth value are called logically
equivalent statements. A conditional and its contrapositive are logically equivalent, and so
are the converse and inverse... see page 83 for more details)
Sec 2-3 Using Deductive Reasoning to Verify Conjectures
To prove that a conjecture is true, you must use DEDUCTIVE REASONING.
14. Deductive reasoning is the process of using l£>Ai^-" _ to draw conclusions from
, and
"'Inductive reasoning is based on PATTERNS from experiences, deductive reasoning is based on LOGIC
based on facts.
Tell whether each conclusion is the result of inductive or deductive reasoning.
15. The United States Census Bureau collects data on the earnings of American citizens. Using data for the
three years from 2001 to 2003, the bureau concluded that the national average median income for a fourperson family was $43,527.
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Indu crfve.
16. A speeding ticket costs $40 plus $5 per mi/h over the speed limit. Lynne mentions to Frank that she was
given a ticket for $75. Frank concludes that Lynne was driving 7 mi/h over the speed limit.
In deductive reasoning, if the given facts are true and you apply the correct logic, then the conclusion
must be true. There are two forms of deductive reasoning:
> Law of Detachment: If /?—» q is a true statement andp is true, then q is true.
> Law of Syllogism: If p—•> q and q —» r are true statements, then p—> q is a true
statement.
Determine if each conjecture is valid/invalid by the Law of Detachment.
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17. Given: IfmZABC = mZC£D,then BC bisects ZA5D. BC bi
Conjecture: m ^ABC = mZ.CBD .
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p is obtuse.
18. Given: An obtuse triangle has^two acute angles. Triangle ABC
Conjecture: Triangle ABC has two acute angles.
Determine if each conjecture is valid/invalid by the Law of Syllogism.
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19. Given: If a figure is a square, then the figure is a rectangle. If a figure is a
Square, then it is a parallelogram.
Conjecture: If a figure is a parallelogram, then it is a rectangle.
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20. Given: If you buy a car, then you can drive to school. If you can drive to school,
then you will not ride the bus.
Conjecture: If you buy a car, then you will not ride the bus.
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Draw a conclusion from the given information using your deductive reasoning skills.
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21. Given: If a polygon is a triangle, then it has three sides. If a polygon has three
sides, then it is not a quadrilateral. Polygon P is a triangle.
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Conclusion: _
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Sec 2-5 Algebraic Proof
Aproofis an argument that uses logic, definitions, properties, and previously proven statements to
show that a conclusion is true.
If you've ever solved an equation, then you've already done a proof! An important part of writing a
proof is giving a justification to show that every step is valid. For each justification, you can use a
definition, postulate, property, or a piece of information that is given. Below is a list of the properties
you will use when writing your algebraic proofs.
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"•OpClTICS Of cCfliuiliy j|gg|BgBBflggg^gBlHiBH^HiHHHI^HBl^B^HH^^B^HflB^
Addition Property of Equality
If a = fa, then a + c = b + c.
Subtraction Property of Equality
If a = b, then a - c = b - c.
Multiplication Property of Equality
If a = b, then ac = be.
Division Property of Equality
If a = fa and c + 0, then f = |.
Reflexive Property of Equality
a=a
Symmetric Property of Equality
If a = fa, then to = a.
Transitive Property of Equality
If a = fa and b - c, then a = c
Substitution Property of Equality
If a = b, then b can be substituted for
a in any expression.
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Identify the property that justifies each statement.
22. DE = GH, so GH = DE.
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23. Zl =
24. Write a justification for each step.
Steps
2p - 30 = -4p + 6
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Justifications (Reasons)
Given
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25. Write a justification for each step for the picture on
the right.
2y+21
B
Steps
AB = BC
5y + 6 = 2y + 21
3y + 6 = 21
3y= 15
V= 5
Justifications (Reasons)
C
26. Use the SEGMENT ADDITION POSTULATE to write your equation,
substitute the given information, and solve the equation. Write a justification
for each step.
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Steps
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Justifications (Reasons)
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Sec 2-6 Geometric proofs
27. If m^l = x + 50 and m/.2 = 3.v - 20, find m£i
28. Find the measure of each numbered angle.
= 5x
9-O
= 4.v - 6
= 12x- - 12 =
A math club decided to buy T-shirts for its members. A clothing company quoted the following prices
for the T-shirts.
Math Club T-Shirts
Numbs rot
Total Cost
(dollars)
T-Shirls
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'5
75
10 . ';,;-,
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15
105
20 . , |
136
29. Write an equation that best describes the relationship between the total cost, c, and the number
of T-shirts, si
30. What would be the total cost of 47 T-shirts?
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31. Compare and contrast Deductive Reasoning and Inductive Reasoning. Then, give an example
when we use each one.
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