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Document 1805457
Chapter 2,
continued
Mixed Review for TAKS
36. a.
0
Class
income
0 0.25 0.50 0.75 1.00 1.25 2.5 5.00
39. D;
1
2
3
4
5
10
20
f(x) 5 mx 1 b
y
6
5
4
3
2
1
0
Each time the value of x increases by 1, the value of f (x)
increases by 5. So, f (x) is a linear function whose rate
of change is 5. Because the function is linear, it can be
written in the form f (x) 5 mx 1 b, where m is the rate of
change. Substitute values from the table to find b.
22 5 5(1) 1 b
27 5 b
So, the expression 5x 2 7 can be used to find the values
of f(x) in the table.
0 5 10 15 20 x
Number of
tickets sold
There is a linear relationship between the number of
tickets sold and the income. The income is 0.25 times
the number of tickets sold.
c. Let y be your income and x be the number of
tickets sold. An equation for your income
is y 5 0.25x.
d. When y 5 14:
14 5 0.25x
56 5 x
Your class must sell more than 56 tickets to
make a profit.
e. To make a profit of $50, your class must sell
14 1 50 5 $64 worth of tickets. Find the value of x
when y 5 64.
64 5 0.25x
256 5 x
Your class must sell 256 tickets to make a profit
of $50.
37. a. After the first two numbers, each number is the sum
of the two previous numbers.
b. 144, 233, 377
c. Sample answer: Spiral patterns on the head of a
sunflower
38. a. Sample answer: A counterexample is 15, which is
a multiple of 5 but not a multiple of 8. So 15 is a
member of set A, but not a member of set B. The
conjecture is false because a counterexample exists.
b. Sample answer: A counterexample is 99, which is
less than 100 but not a member of set A or set B. The
conjecture is false because a counterexample exists.
c. A counterexample is 40, which is in both set A and
set B. The conjecture is false because a
counterexample exists.
40. H;
V 5 Bh
5 :r 2h
ø 3.14(5.52)6.2
5 3.14(30.25)6.2
ø 589
The approximate volume of the cylinder is
589 cubic centimeters.
Lesson 2.2
2.2 Guided Practice (pp. 79–82)
1. If an angle is a 908 angle, then it is a right angle.
2. If x 5 23, then 2x 1 7 5 1.
3. If n 5 9, then n 2 5 81.
4. If a tourist is at the Alamo, then the tourist is in Texas.
5. Converse: If a dog is large, then it is a Great Dane.
False, not all large dogs are Great Danes.
Inverse: If a dog is not a Great Dane, then it is not large.
False, a dog could be large but not a Great Dane.
Contrapositive: If a dog is not large, then it is not a
Great Dane.
True, a dog that is not large cannot be a Great Dane.
6. Converse: If a polygon is regular, then the polygon
is equilateral.
True, all regular polygons are equilateral.
Inverse: If a polygon is not equilateral, then it is
not regular.
True, a polygon that is not equilateral cannot be regular.
Contrapositive: If a polygon is not regular, then the
polygon is not equilateral.
False, a polygon that is not regular can still be equilateral.
7. True. Ž JMF and Ž FMG form a linear pair so they are
supplementary.
30
Geometry
Worked-Out Solution Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
b.
Total income (dollars)
Number
of
tickets
sold
Chapter 2,
continued
}
8. False. It is not known that M bisects FH. So, you cannot
}
state that M is the midpoint for FH.
9. True. Ž JMF and Ž HMG are vertical angles because
11. False; a polygon can have 5 sides without being a regular
pentagon.
Counterexample:
their sides form two pairs of opposite rays.
10. False. It is not shown that @##$
FH and @##$
JG intersect to form
@##$ @##$
right angles. So, you cannot state that FH>
JG .
11. An angle is a right angle if and only if it measures 908.
12. True.
13. False; two angles can be supplementary without being a
linear pair. Counterexample:
12. Mary will be in the fall play if and only if she is in
theater class.
2.2 Exercises (pp. 82–85)
1358
B
458
C
E
D
14. True.
Skill Practice
1. The converse of a conditional statement is found by
switching the hypothesis and the conclusion.
2. Collinear points are points that lie on the same line.
Points are collinear if and only if they lie on the
same line.
3. If x 5 6, then x 5 36.
2
4. If an angle is a straight angle, then it measures 1808.
5. If a person is registered to vote, then that person is
allowed to vote.
6. The error is in identifying the correct hypothesis and
15. False; Counterexample: The number 5 is real, but
not irrational.
16. True; Ž ABC is a right angle, so mŽ ABC 5 908.
17. False; It is not known that Ž1 is a right angle, so you
PQ > @##$
ST .
cannot conclude that @##$
18. True; Ž2 and Ž3 are adjacent angles whose noncommon
sides form opposite rays, so Ž2 and Ž3 are a linear
pair. Angles in a linear pair are supplementary,
so mŽ2 1 mŽ3 5 1808.
19. An angle is obtuse if and only if its measure is
between 908 and 1808.
conclusion when writing the if-then form of the statement.
The hypothesis is “a student is in high school” and the
conclusion is “the student takes four English courses.”
20. Two angles are a linear pair if and only if they are
If-then statement: If a student is in high school, then the
student takes four English courses.
21. Points are coplanar if and only if they lie in the
7. If-then: If two angles are complementary, then they
add to 908.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
F
A
Converse: If two angles add to 908, then they
are complementary.
Inverse: If two angles are not complementary, then they
do not add to 908.
Contrapositive: If two angles do not add to 908, then they
are not complementary.
8. If-then: If an animal is an ant, then it is an insect.
Converse: If an animal is an insect, then it is an ant.
Inverse: If an animal is not an ant, then it is not an insect.
Contrapositive: If an animal is not an insect, then it is not
an ant.
9. If-then: If x 5 2, then 3x 1 10 5 16.
Converse: If 3x 1 10 5 16, then x 5 2.
adjacent angles whose noncommon sides are
opposite rays.
same plane.
22. This is not a valid definition. The converse of the
statement is not true. Rays can have a common endpoint
without being opposite rays.
23. The statement is a valid definition.
24. The statement is not a valid definition. The converse
of the statement is false. If the measure of an angle
is greater than that of an acute angle, the angle is not
necessarily
a right angle.
25. A; If you do your homework, then you can go to the
movie afterwards. This is the if-then form of the
given statement.
26. If x > 0, then x > 4. A counterexample is x 5 2. Note
that 2 > 0, but 2 ò 4. Because a counterexample exists,
the converse is false.
Inverse: If x Þ 2, then 3x 1 10 Þ 16.
27. If 2x > 26, then x < 6. The converse is true.
Contrapositive: If 3x 1 10 Þ 16, then x Þ 2.
28. If xa0, then xa2x. The converse is true.
10. If-then: If a point is a midpoint, then it bisects a segment.
Converse: If a point bisects a segment, then it is
a midpoint.
Inverse: If a point is not a midpoint, then it does not
bisect a segment.
29. Sample answer: If x 5 2, then x 2 > 0.
30. If Ž1 and Ž2 are linear pairs, then mŽ2 is 908; if Ž1 and
Ž4 are linear pairs, then mŽ4 is 908; if Ž4 and Ž3 are
linear pairs, then mŽ3 is 908.
Contrapositive: If a point does not bisect a segment, then
it is not a midpoint.
Geometry
Worked-Out Solution Key
31
continued
Problem Solving
c. Sample answer: If it is a rock, then it can be formed
in different ways. The converse of the statement is
false. If something can be formed in different ways, it
doesn’t necessarily mean it has to be a rock. It could
be soil for example.
31. Statement: If a fragment has a diameter greater than 64
millimeters, then it is called a block or bomb.
Converse: If a fragment is called a block or bomb, then
it has a diameter greater than 64 millimeters.
Both the statement and its converse are true. So, the
biconditional statement is true.
32. Counterexample: a fragment with a diameter of 1
millimeter
The diameter is less than 64 millimeters, but the fragment
is not called a lapilli. Because a counterexample exists,
the biconditional statement is false.
33. You can show that the statement is false by finding a
counterexample. Some sports do not require helmets,
such as swimming or track.
34. a. The statement is true. The mean is the average value
of the data, so it will lie between the least and greatest
values in the data set.
b. If the mean of your data set is between x and y, then x
and y are the least and greatest values in your data set.
The converse is false. The mean is between any two
numbers in a data set where one of the numbers is less
than the mean and the other is greater than the mean.
The numbers do not have to be the least and greatest
values in the data set.
37. The statement cannot be written as a true biconditional.
The biconditional is false because x 5 23 also makes
the statement true. A counterexample exists, so the
biconditional statement is false.
38. For a statement to be a true biconditional, both the
original statement and the converse must be true. If the
contrapositive of a statement is true, then you know that
the original statement is true. However, you do not know
if the converse is true. So, you don’t know if it can be
written as a true biconditional.
39. It is Tuesday. Because it is Tuesday, I have art class.
Because I have art class, I do not have study hall.
Because I do not have study hall, I must have
music class.
Mixed Review for TAKS
40. A;
Number of successes
Experimental probability 5 }}
Number of trials
4
5}
25
5 0.16
c. If a data set has a mean, median, and mode, then
the mode of the data set will always be one of the
measurements.
The mode is the data value that occurs most frequently
in a data set. So, if the mode exists, then it will always
be one of the data values. The median is one of the
data values only when there is an odd number of
values in the data set. The mean does not have to be a
data value.
The experimental probability that the spinner lands on
red is 0.16.
41. H;
@##$ is not shown in the figure.
XW
Lesson 2.3
Investigating Geometry Activity 2.3 (p. 86)
n-dimensional geometry
Differential calculus
Math for theory of relativity
Perspective drawing
Pythagorean Theorem
Did not eat beans
Studied moonlight
Wrote a math book at 17
Fluent in Latin
Played piano
35. Sample answer: If a student is in the jazz band, then the
student is in the band.
36. a. If a rock is formed from the cooling of molten rock,
then it is igneous rock.
If a rock is formed from pieces of other rocks, then it
is sedimentary rock.
If a rock is formed by changing temperature, pressure,
or chemistry, then it is metamorphic rock.
b. If a rock is igneous rock, then it is formed from the
cooling of molten rock.
If a rock is sedimentary, then it is formed from pieces
of other rocks.
If a rock is metamorphic, then it is formed by
changing temperature, pressure, or chemistry.
The converse of each statement is true.
If a rock is classified in one of these ways, it must be
formed in the manner described.
32
Geometry
Worked-Out Solution Key
Maria Agnesi
Anaxagoras
Emmy Noether
Julio Rey Pastor
Pythagoras
Did not eat beans
Studied moonlight
Wrote a math book at 17
Fluent in Latin
Played piano
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Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Chapter 2,
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