Download Lesson 2.1 • Inductive Reasoning

Lesson 2.1 • Inductive Reasoning
Name
Period
Date
For Exercises 1–8, use inductive reasoning to find the next two terms in
each sequence.
1. 4, 8, 12, 16, _____, _____
2. 400, 200, 100, 50, 25, _____, _____
1 2 1 4
8 7 2 5
4. 5, 3, 2, 1, 1, 0, _____, _____
3. , , , , _____, _____
5. 360, 180, 120, 90, _____, _____
6. 1, 3, 9, 27, 81, _____, _____
7. 1, 5, 17, 53, 161, _____, _____
8. 1, 5, 14, 30, 55, _____, _____
For Exercises 9–12, use inductive reasoning to draw the next two shapes in
each picture pattern.
9.
10.
11.
12.
y
y
y
(–1, 3)
(3, 1)
x
x
x
(–3, –1)
For Exercises 13–15, use inductive reasoning to test each conjecture.
Decide if the conjecture seems true or false. If it seems false, give
a counterexample.
13. Every odd whole number can be written as the difference of
two squares.
14. Every whole number greater than 1 can be written as the sum of
two prime numbers.
15. The square of a number is larger than the number.
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CHAPTER 2
9
Lesson 2.2 • Deductive Reasoning
Name
Period
Date
1. ABC is equilateral. Is ABD equilateral? What type of reasoning,
B
inductive or deductive, do you use when solving this problem?
1
2. If 6 # 8 7, 10 # 3 6, and 3 # 2 2.5, then
2
4 # 8 _____
5 # 0 _____
2 # 2 _____
C
D
A
What type of reasoning, inductive or deductive, do you use when
solving this problem?
3. A and D are complementary. A and E are supplementary.
What can you conclude about D and E? What type of reasoning,
inductive or deductive, do you use when solving this problem?
4.
g.
a.
d.
e.
b.
f.
c.
Whatnots
Not whatnots
Which are whatnots?
What type of reasoning, inductive or deductive, do you use
when solving this problem?
5. Solve each equation for x. Give a reason for each step in the process.
19 2(3x 1)
5
What type of reasoning, inductive or deductive, do you use when
solving these problems?
a. 4x 3(2 x) 8 2x
b. x 2
6. A sequence is generated by the function f(n) 5 n 2. Give the first
five terms in the sequence. What type of reasoning, inductive or
deductive, do you use when solving this problem?
7. A sequence begins 4, 1, 6, 11 . . .
a. Give the next two terms in the sequence. What type of reasoning,
inductive or deductive, do you use when solving this problem?
b. Find a rule that generates the sequence. Then give the 50th term in
the sequence. What type of reasoning, inductive or deductive, do
you use when solving this problem?
8. Choose any 3-digit number. Multiply it by 7. Multiply the result by 11.
Then multiply the result by 13. Do you notice anything? Try a few
other 3-digit numbers and make a conjecture. Use deductive reasoning
to explain why your conjecture is true.
10
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Lesson 2.3 • Finding the nth Term
Name
Period
Date
For Exercises 1–4, tell whether or not the rule is a linear function.
1.
n
1
2
3
4
5
f(n)
8
15
22
29
36
3. n
h(n)
2.
n
g(n)
1
2
3
4
5
9
6
2
3
9
4.
1
2
3
4
5
14
11
8
5
2
1
2
n
j(n)
3
4
5
32 1 12
0
1
2
For Exercises 5 and 6, complete each table.
5. n
1
2
3
4
5
6. n
6
f(n) 7n 12
1
2
3
4
5
6
g(n) 8n 2
For Exercises 7–9, find the function rule for each sequence. Then find the
50th term in the sequence.
7. n
f(n)
8. n
g(n)
9. n
h(n)
1
2
3
4
5
6
9
13
17
21
25
29
1
2
3
4
5
6
6
1
4
9
1
2
3
4
5
6
6.5
7
7.5
8
8.5
9
...
n
...
50
...
n
...
50
...
n
...
50
14 19
10. Find the rule for the number of tiles in the nth figure. Then find the
number of tiles in the 200th figure.
n
1
2
3
Number
of tiles
1
4
7
4
5
...
n
...
200
11. Sketch the next figure in the sequence. Then complete the table.
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n
1
2
Number of
segments
and lines
2
6
Number of
regions of
the plane
4
3
4
...
n
...
CHAPTER 2
50
11
Lesson 2.4 • Mathematical Modeling
Name
Period
Date
1. If you toss a coin, you will get a head or a tail. Copy and complete the
geometric model to show all possible results of four consecutive tosses.
H
H
T
How many sequences of results are possible? How many sequences have
exactly one tail? Assuming a head or a tail is equally likely, what is the
probability of getting exactly one tail in four tosses?
2. If there are 12 people sitting around a table, how many different pairs
of people can have conversations during dinner, assuming they can
all talk to each other? What geometric figure can you use to model
this situation?
3. Tournament games and results are often displayed using a geometric
model. Two examples are shown below. Sketch a geometric model for
a tournament involving 4 teams and a tournament involving 5 teams.
Each team must have the same chance to win. Try to have as few
games as possible in each tournament. Show the total number
of games in each tournament. Name the teams a, b, c . . . and number
the games 1, 2, 3 . . . .
a
a
1
b
3
1
3
c
b
2
3 teams, 3 games
(round robin)
12
CHAPTER 2
c
d
2
4 teams, 3 games
(single elimination)
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Lesson 2.5 • Angle Relationships
Name
Period
Date
For Exercises 1–8, find each lettered angle measure without using
a protractor.
1.
2.
a
112°
3.
a
15°
b c
4.
e 132°
d
6.
b
7.
c
c
a
b
c
8.
a
b
ba
42°
d
a
70°
5.
66°
40°
38°
c
70°
d e
110° a
b
138°
100°
a
b
25°
For Exercises 9–14, tell whether each statement is always (A),
sometimes (S), or never (N) true.
9. _____ The sum of the measures of two acute angles equals the
measure of an obtuse angle.
10. _____ If XAY and PAQ are vertical angles, then either X, A, and P
or X, A, and Q are collinear.
11. _____ The sum of the measures of two obtuse angles equals the
measure of an obtuse angle.
12. _____ The difference between the measures of the supplement and the
complement of an angle is 90°.
13. _____ If two angles form a linear pair, then they are complementary.
14. _____ If a statement is true, then its converse is true.
For Exercises 15–19, fill in each blank to make a true statement.
15. If one angle of a linear pair is obtuse, then the other is ____________.
16. If A B and the supplement of B has measure 22°, then
mA ________________.
17. If P is a right angle and P and Q form a linear pair, then
mQ is ________________.
18. If S and T are complementary and T and U are supplementary,
then U is a(n) ________________ angle.
19. Switching the “if ” and “then” parts of a statement changes the
statement to its ________________.
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CHAPTER 2
13
Lesson 2.6 • Special Angles on Parallel Lines
Name
Period
Date
For Exercises 1–11, use the figure at right.
For Exercises 1–5, find an example of each term.
1. Corresponding angles
2. Alternate interior angles
3. Alternate exterior angles
4. Vertical angles
3 4
7 8
1 2
5 6
5. Linear pair of angles
For Exercises 6–11, tell whether each statement is always (A), sometimes
(S), or never (N) true.
6. _____ 1 3
7. _____ 3 8
8. _____ 2 and 6 are supplementary.
9. _____ 7 and 8 are supplementary.
10. _____ m1 m6
11. _____ m5 m4
For Exercises 12–14, use your conjectures to find each angle measure.
12.
13.
14.
54°
a
a
a
65°
b
b
54°
c
d
c
b
For Exercises 15–17, use your conjectures to determine whether or not
1 2, and explain why. If not enough information is given, write “cannot
be determined.”
15.
16.
118°
62°
17.
1
1
2
18. Find each angle measure.
48°
95°
48°
2
25°
1
2
44°
f
78°
e
64°
d
c
a
b
14
CHAPTER 2
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LESSON 1.7 • A Picture Is Worth a Thousand Words
1.
8. Possible answer:
9. Possible answer:
Possible locations
Gas
5
10 m
A
12 m
5
Power
2.
30 m
11. x 2, y 1
10. 18 cubes
Wall
4m
12 m
Station 1
LESSON 2.1 • Inductive Reasoning
Station 2
12 m
Possible locations
4m
Wall
3. Dora, Ellen, Charles, Anica, Fred, Bruce
1
1
3
1. 20, 24
1 1
2. 122, 64
5
3. 4, 2
4. 1, 1
5. 72, 60
6. 243, 729
7. 485, 1457
8. 91, 140
9.
4
3
10.
D E C A F
11.
B
4. Possible answers:
a.
b.
c.
12.
y
y
(3, 1)
x
x
(1, –3)
13. True
LESSON 1.8 • Space Geometry
1.
2.
14. False; 11 is not a sum of any two prime numbers.
1 2 1
15. False; 2 4
LESSON 2.2 • Deductive Reasoning
3.
2. 6, 2.5, 2; inductive
1. No; deductive
3. mE mD (mE mD 90°); deductive
4. a, e, f; inductive
5. Deductive
4. Rectangular prism
5. Pentagonal prism
a. 4x 3(2 x) 8 2x
4x 6 3x 8 2x
x 6 8 2x
3x 6 8
6.
7.
3x 2
2
x 3
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The original equation.
Distributive property.
Combining like terms.
Addition property of
equality.
Subtraction property of
equality.
Division property of
equality.
ANSWERS
89
19 2(3x 1)
b. 5 x 2 The original equation.
19 2(3x 1) 5(x 2) Multiplication property
11. (See table at bottom of page.)
of equality.
19 6x 2 5x 10 Distributive property.
21 6x 5x 10 Combining like terms.
21 11x 10 Addition property of
equality.
11 11x
LESSON 2.4 • Mathematical Modeling
Subtraction property of
equality.
1x
1.
H
Division property of
equality.
H
T
6. 4, 1, 4, 11, 20; deductive
H
H
7. a. 16, 21; inductive
b. f(n) 5n 9; 241; deductive
T
T
8. Sample answer: If any 3-digit number “XYZ” is
multiplied by 7 11 13, then the result will
be of the form “XYZ,XYZ.” This is because
7 11 13 1001. For example,
451
H
H
T
T
7 11 13 451(7 11 13)
H
T
451(1001)
T
451(1000 1)
LESSON 2.3 • Finding the nth Term
2. Linear
T
H
T
H
H
T
H
T
H
T
H
T
H
T
3. Not linear
3. Possible answers:
b
4. Linear
5.
n
1
1
f (n)
2
5
3
2
4
9
5
16
6
23
30
1
2
3
4
5
6
g(n)
10
18
26
34
42
50
3
4
4
5
6
f
6 teams, 7 games
5 teams, 10 games
a
b
1
8. f(n) 5n 11; f(50) 239
1
9. f(n) 2n 6; f(50) 31
10.
2
7
d
4
e
7. f(n) 4n 5; f(50) 205
1
c
3
e
d
8
10
5
2
a
7 3
9
1 6
n
n
c
2
b
a
6.
Sequences with
exactly one tail
T
16 sequences of results. 4 sequences have exactly
4
1
one tail. So, P(one tail) 1
64
2. 66 different pairs. Use a dodecagon showing sides
and diagonals.
451,000 451 451,451
1. Linear
H
f
6
4
3
5
e
2
5
n
200
c
d
Number
of tiles
1
4
7
10
13
3n 2
6 teams, 6 games
598
Lesson 2.3, Exercise 11
11.
90
Figure number
1
2
3
4
n
50
Number of segments and lines
2
6
10
14
4n 2
198
Number of regions of the plane
4
12
20
28
8n 4
396
ANSWERS
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LESSON 2.5 • Angle Relationships
LESSON 3.1 • Duplicating Segments and Angles
1. a 68°, b 112°, c 68°
2. a 127°
1.
P
Q
3. a 35°, b 40°, c 35°, d 70°
R
S
4. a 24°, b 48°
5. a 90°, b 90°, c 42°, d 48°, e 132°
A
6. a 20°, b 70°, c 20°, d 70°, e 110°
7. a 70°, b 55°, c 25°
8. a 90°, b 90°
9. Sometimes
10. Always
11. Never
13. Never
14. Sometimes 15. acute
16. 158°
17. 90°
B
2. XY 3PQ 2RS
X
Y
12. Always
18. obtuse
3. Possible answer:
128° 35° 93°
4.
B
19. converse
LESSON 2.6 • Special Angles on Parallel Lines
C
1. One of: 1 and 3; 5 and 7; 2 and 4;
6 and 8
5.
6.
B
D
D
2. One of: 2 and 7; 3 and 6
3. One of: 1 and 8; 4 and 5
C
4. One of: 1 and 6; 3 and 8; 2 and 5;
4 and 7
5. One of: 1 and 2; 3 and 4; 5 and 6;
7 and 8; 1 and 5; 2 and 6; 3 and 7;
4 and 8
6. Sometimes
9. Always
7. Always
10. Never
D
D
C
7. Four possible triangles. One is shown below.
8. Always
11. Sometimes
12. a 54°, b 54°, c 54°
13. a 115°, b 65°, c 115°, d 65°
14. a 72°, b 126°
16. 1 2
15. 1 2
17. cannot be determined
18. a 102°, b 78°, c 58°, d 122°, e 26°,
f 58°
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LESSON 3.2 • Constructing Perpendicular Bisectors
1.
2. Square
ANSWERS
91