Chapter 2 – Reasoning and ProofAnswer Key
2.1
Conjectures and Counterexamples
Answers
1.
True
2.
COULD NOT IDENTIFY THE CONJECTURE
3.
False, maybe a raccoon ate the bread with peanut butter instead of the chipmunk
4.
False, it is possible that the other students, that are not her friends, in her geometry class are
at different grade levels.
5.
Answers vary
6.
n = 1 would be a counterexample because 12 ≯1 . Recall that a whole number is
0, 1, 2, 3, … n, so 0 could also be a counterexample.
7.
8.
Counterexamples include: 21, 51, 81, 121, and 151
is one counterexample. Any positive improper fraction (where the numerator is greater
than the denominator) could be a counterexample.
9.
A triangle is a counterexample.
10. A girl that doesn’t like ice cream would be a counterexample.
CK­12 Geometry Concepts
1
Chapter 2 – Reasoning and ProofAnswer Key
11. Not everyone takes choir in high school.
12. Obtuse angles do not have complementary angles.
13. 13, 14, 15 year­olds are teenagers and cannot drive yet.
14. Any negative integer could be a counterexample.
15. Equations can have any real number as a solution.
CK­12 Geometry Concepts
2
Chapter 2 – Reasoning and ProofAnswer Key
2.2
If­Then Statements
Answers
1.
Hypothesis: 5 divides evenly into x.
Conclusion: x ends in 0 or 5.
2.
Hypothesis: A triangle has three congruent sides.
Conclusion: It is an equilateral triangle.
3.
Here, the “if” is in the middle of the statement, making the hypothesis the second half.
Hypothesis: Three points lie in the same plane.
Conclusion: The three points are coplanar.
4.
Hypothesis: x = 3.
Conclusion:
5.
.
Hypothesis: You take yoga.
Conclusion: You are relaxed.
6.
Hypothesis: You are a baseball player.
Conclusion: You wear a hat.
7.
Hypothesis: I am 16 years old.
Conclusion: I will learn how to drive.
CK­12 Geometry Concepts
3
Chapter 2 – Reasoning and ProofAnswer Key
8.
Hypothesis: You do your homework.
Conclusion: You can watch TV.
9.
Hypothesis: The lines are parallel.
Conclusion: Alternate interior angles are congruent.
10.
Hypothesis: You are a kid.
Conclusion: You like ice cream.
11.
If it is Thursday, then Susie is eating pizza.
12.
If you are Raychel, then you completed your homework
13.
If it is a weekday, then Alex is going to school
14.
If you are a student, then you have a math class.
15.
If a shape is a square, then it has right angles
CK­12 Geometry Concepts
4
Chapter 2 – Reasoning and ProofAnswer Key
2.3
Converse, Inverse, and Contrapositive
Answers
1.
If B is the midpoint of
, then AB = 5 and BC = 5. This could be true, but we don’t
know the length of AC. AB = BC, but we cannot say they are 5 without knowing the
length of AC.
2.
If AB
5 and BC
5, then B is not the midpoint of
. Again, this could be true, but
we don’t know AC. Also, A, B and C might not be collinear.
3.
A counterexample would be if the points A, B, and C are not collinear, thus not creating
the line segment
5 and BC
.
4.
If AB
5, then B is not the midpoint of
5.
p→q
6.
Contrapositive
7.
Contrapositive
8.
p→q
9.
If a U.S. citizen can vote, then he or she is 18 or more years old.
CK­12 Geometry Concepts
.
5
Chapter 2 – Reasoning and ProofAnswer Key
If he or she is 18 or more years old, then he or she is a U.S. citizen that can vote
10.
If a whole number is prime, then it has exactly two distinct factors.
If a whole number has exactly two distinct factors, then it is prime
11.
If the points are collinear, then there is a line that contains the points.
If the points are on the same line, then they are collinear.
12.
If 2x = 18 , then x = 9 .
If x = 9 , then 2x = 18 .
13.
14.
15.
a)
True
b)
False, x = 4 OR x =− 4
c)
False, let x =− 4 , then x 2 = 16
d)
True
a)
True
b)
True
c)
True
d)
True
a)
True
b)
True
c)
True
CK­12 Geometry Concepts
6
Chapter 2 – Reasoning and ProofAnswer Key
d)
True
CK­12 Geometry Concepts
7
Chapter 2 – Reasoning and ProofAnswer Key
16.
a)
True
b)
False, Any angle that is less than 90° is an acute angle.
c)
False, Any angle that is less than 90° is an acute angle.
d)
True
17.
Answers vary
18.
Answers vary
CK­12 Geometry Concepts
8
Chapter 2 – Reasoning and ProofAnswer Key
2.4
Inductive Reasoning from Patterns
Answers
1.
4th figure: 9 dots, 10th figure: 21dots
2.
4th figure: 20 dots, 10th figure: 110 dots
3.
a)
4.
5.
b)
There are two more points in each star than its figure number.
a)
10 triangles
b)
48
c)
2n , for any positive integer n .
1)
20, 23
2)
107
CK­12 Geometry Concepts
9
Chapter 2 – Reasoning and ProofAnswer Key
3)
3n + 2 , for any positive integer n .
CK­12 Geometry Concepts
10
Chapter 2 – Reasoning and ProofAnswer Key
6.
7.
8.
9.
10.
1)
­19, ­24
2)
­164
3)
11 − 5n , for any positive integer n .
1)
64, 128
2)
34,359,738,370
3)
2n , for any positive integer n .
1)
12, 1
2)
­307
3)
78 − 11n , for any positive integer n.
1)
6 7
7, 8.
2)
35
36
3)
n
n+1
1)
12 14
23 , 27
2)
70
139
3)
2n
4n−1
, for any positive integer n.
.
.
.
CK­12 Geometry Concepts
11
Chapter 2 – Reasoning and ProofAnswer Key
11.
12.
13.
14.
1)
36, 49
2)
1225
3)
n2 , for any positive integer n.
1)
20, 27
2)
860
3)
n(n+3)
2
1)
21, 28
2)
861
, for any positive integer n.
3)
(n+1)(n+2)
2
1)
105, 168
2)
34440
3)
, for any positive integer n.
n(n+1)(n+2)
2
, for any positive integer n.
CK­12 Geometry Concepts
12
Chapter 2 – Reasoning and ProofAnswer Key
15.
The points are collinear. The equation of this graph is y = 5x − 2
16.
Answers vary.
CK­12 Geometry Concepts
13
Chapter 2 – Reasoning and ProofAnswer Key
2.5
Deductive Reasoning
Answers
1.
Law of Detachment
2.
No conclusion
3.
Law of Syllogism.
4.
Law of Contrapositive.
5.
Law of Detachment.
6.
Law of Contrapositive
7.
Law of Contrapositive.
8.
Law of Syllogism
9.
This is a logical argument, but it doesn’t make sense because we know that circles exist.
CK­12 Geometry Concepts
14
Chapter 2 – Reasoning and ProofAnswer Key
CK­12 Geometry Concepts
15
Chapter 2 – Reasoning and ProofAnswer Key
10.
p→q
p
∴q
11.
p→q
˜q
∴ ˜p
12.
If I need a new watch battery, then I go to the mall. If I don’t shop, then I didn’t go to the
mall, but since I went to the mall, then I shop. If I shop, then I will buy shoes. Since I shop,
then I conclude that I bought shoes.
13.
If Anna’s teacher gives, notes, Anna writes them down. If Anna writes down the notes, she
can do the homework. If Anna’s parents don’t buy her ice cream, then she didn’t get an A
on her test. If Anna didn’t get an A on her test, then she couldn’t do the homework. But
since she did do her homework, she got an A on her test. Since she got an A on her test,
we conclude that her parents bought her ice cream.
14.
Inductive Reasoning
15.
Deductive Reasoning
16.
Inductive Reasoning
17.
Inductive Reasoning
CK­12 Geometry Concepts
16
Chapter 2 – Reasoning and ProofAnswer Key
CK­12 Geometry Concepts
17
Chapter 2 – Reasoning and ProofAnswer Key
18.
Inductive Reasoning
19.
Deductive Reasoning
20.
Deductive Reasoning
CK­12 Geometry Concepts
18
Chapter 2 – Reasoning and ProofAnswer Key
2.6
Truth Tables
Answers
1.
p
T
T
T
T
F
F
F
F
q
r
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
q
r
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
q
r
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
˜r
F
T
F
T
F
T
F
T
p∧q
T
T
F
F
F
F
F
F
(p∧q)∨ ˜r
T
T
F
T
F
T
F
T
˜q
F
F
T
T
F
F
T
T
˜q∨r
T
F
T
T
T
F
T
T
p∨( ˜q∨r)
T
T
T
T
T
F
T
T
˜r
F
T
F
T
F
T
F
T
q∨ ˜r
T
T
F
T
T
T
F
T
p∧(q∨ ˜r)
T
T
F
T
F
F
F
F
2.
p
T
T
T
T
F
F
F
F
3.
p
T
T
T
T
F
F
F
F
CK­12 Geometry Concepts
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Chapter 2 – Reasoning and ProofAnswer Key
CK­12 Geometry Concepts
20
Chapter 2 – Reasoning and ProofAnswer Key
4.
There are a lot more false cells in #3 than #1
5.
When at least one p, q, or r is true.
6.
q
T
T
F
F
T
T
F
F
p
T
T
T
T
F
F
F
F
p∨q∨r
T
T
T
T
T
T
T
F
r
T
F
T
F
T
F
T
F
7.
p
T
T
T
T
F
F
F
F
q
r
p∨q
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
T
T
F
F
˜r
F
T
F
T
F
T
F
T
(p∨q)∨ ˜r
T
T
T
T
T
T
F
T
8.
p
T
T
T
T
F
F
q
r
T
T
F
F
T
T
T
F
T
F
T
F
CK­12 Geometry Concepts
˜p
F
F
F
F
T
T
˜q
F
F
T
T
F
F
˜p∧ ˜q
F
F
F
F
F
F
( ˜p∧ ˜q)∧r
F
F
F
F
F
F
21
Chapter 2 – Reasoning and ProofAnswer Key
F
F
F
F
T
F
CK­12 Geometry Concepts
T
T
T
T
T
T
T
F
22
Chapter 2 – Reasoning and ProofAnswer Key
9.
p
T
T
T
T
F
F
F
F
10.
True
11.
False
12.
True
13.
False
14.
True
15.
False
q
r
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
CK­12 Geometry Concepts
˜p
F
F
F
F
T
T
T
T
˜q
F
F
T
T
F
F
T
T
˜p∨ ˜q
F
F
T
T
T
T
T
T
( ˜p∨ ˜q)∧r
F
F
T
F
T
F
T
F
23
Chapter 2 – Reasoning and ProofAnswer Key
2.7
Properties of Equality and Congruence
Answers
1.
3x + 11 = ­16
3x = ­27
x = ­9
2.
Addition PoE
x = ­8
Division PoE
g + 1 = 19
g = 27
Subtraction PoE
Multiplication PoE
MN = 5
MN = 10
5.
Subtraction PoE
4x = ­32
g = 18
4.
Division PoE (divide both sides by 3)
7x – 3 = 3x – 35
4x – 3 = ­35
3.
Subtraction PoE (subtract 11 from both sides)
Multiplication PoE
5m∠ABC = 540°
m∠ABC = 108°
CK­12 Geometry Concepts
Division PoE
24
Chapter 2 – Reasoning and ProofAnswer Key
6.
10b – 2(b + 3) = 5b
10b – 2b + 6 = 5b
8b + 6 = 5b
Distributive Property
Combine like terms
6 = ­3b
Subtraction PoE
­2 = b
Division PoE
b = ­2
Symmetric PoE (this step is not necessary, but helpful to
see how the Symmetric property works)
7.
Multiplication PoE (multiplied everything by 12)
3y = ­6
y = ­2
Subtraction PoE
Division PoE
*Students could have also found a common denominator and would have ended up with
the same answer.*
8.
3AB + 4 AB = 144 + 6AB
7AB = 144 + 6AB
AB = 144
Multiplication PoE (multiplied everything by 12)
Combine like terms
Subtraction PoE
9.
y + z = x +y
10.
CD = 5
11.
Substitute y – 7 for x in the second equation: y – 7 = z + 4
12.
6x – 12 = y
13.
Yes, they are collinear.
CK­12 Geometry Concepts
25
Chapter 2 – Reasoning and ProofAnswer Key
14.
No, they are not collinear. Since the points are not collinear then there exist no midpoint
because no line segment exists that contains those points.
15.
Since the m∠N OP = 124°, then that angle has to be obtuse since all obtuse angles have
an angle measurement greater than 180°.
CK­12 Geometry Concepts
26
Chapter 2 – Reasoning and ProofAnswer Key
2.8
Two­Column Proofs
Answers
1.
Statement
1. ∠MLN ≅ ∠OLP
2. m∠MLN = m∠OLP
3. m∠MLO = m∠MLN + m∠NLO
m∠NLP = m∠NLO + m∠OLP
4. m∠NLP = m∠NLO + m∠MLN
5. m∠MLO = m∠NLP
6. ∠NLP ≅ ∠MLO
Reason
Given
≅ angles have = measures
Angle Addition Postulate
Substitution
Substitution
≅ angles have = measures
2.
Statement
1.
,
2. ∠BED is a right angle
∠AEC is a right angle
3. m∠BED = 90°
m∠AEC = 90°
4. m∠BED = m∠2 + m∠3
m∠AEC = m∠1 + m∠3
5. 90° = m∠2 + m∠3
90° = m∠1 + m∠3
6. m∠2 + m∠3 = m∠1 + m∠3
7. m∠2 = m∠1
8. ∠2 ≅ ∠1
CK­12 Geometry Concepts
Reason
Given
⊥ lines create right angles
Definition of a right angle
Angle Addition Postulate
Substitution
Substitution
Subtraction PoE
≅ angles have = measures
27
Chapter 2 – Reasoning and ProofAnswer Key
3.
Statement
1. ∠1 ≅ ∠4
2. m∠1 = m∠4
3. ∠1 and ∠2 are a linear pair
∠3 and ∠4 are a linear pair
4. ∠1 and ∠2 are supplementary
∠3 and ∠4 are supplementary
5. m∠1 + m∠2 = 180°
m∠3 + m∠4 = 180°
6. m∠1 + m∠2 = m∠3 + m∠4
7. m∠1 + m∠2 = m∠3 + m∠1
8. m∠2 = m∠3
9. ∠2 ≅ ∠3
Reason
Given
≅ angles have = measures
Given (by looking at the picture) could also
be Definition of a Linear Pair
Linear Pair Postulate
Definition of supplementary angles
Substitution
Substitution
Subtraction PoE
≅ angles have = measures
4.
Statement
1.
2. ∠1 and ∠2 are right angles
3. ∠1 ≅ ∠2
Reason
Given
⊥ lines create right angles.
Right Angles Theorem
5.
Statement
1.
2. ∠1 and ∠2 make a right angle
3. m∠1 + m∠2 = 90°
4. ∠1 and ∠2 are complementary
Reason
Given
⊥ lines create right angles
Definition of a right angle OR
Substitution PoE
Definition of complementary angles
ANSWERS MAY VARY FOR PROBLEMS 6 ­ 13
6.
∠AH G and ∠CH E , ∠GH E and ∠AH C
7.
AH ≅ H E , GH ≅ H C , MH ≅ H P , GM ≅ MA , EP ≅ P C
8.
∠P MA and ∠P MG , ∠MP E and ∠MP C .
CK­12 Geometry Concepts
28
Chapter 2 – Reasoning and ProofAnswer Key
9.
∠AH C
10.
∠MH A and ∠AH C , ∠GH E and ∠EH P
11.
GC
12.
AE and GC
13.
∠GH M and ∠AH M
14.
∠AGC and ∠EGC
15.
NOT COMPLETE
Statement
∠
ACH
≅∠ECH
1.
. CH is on the interior of ∠ACE
1. Given
2. m∠ACH = m∠ECH
3.
4.
5. m∠ACE = 2m∠ACH
6.
7.
2.
3. Angle Addition Postulate
4. Substitution
5.
6. Division PoE
7.
CK­12 Geometry Concepts
Reason
29