Download Math 20 Reasoning and Geometry Review

Math 20
Reasoning and Geometry Review
1.
Suppose you draw some triangle and measure the angles of the interior angles and 1
exterior angle. Write a conjecture about the relationship between the exterior angle
of a triangle and the sum of the two interior opposite angles.
2.
Suppose you draw some isosceles triangles and measure the angles opposite the
equal sides.
The angles opposite the equal sides are equal. What conjecture might you make?
3.
Suppose that you have the following pattern. What conjecture might you make?
2 2  12  3
32  2 2  5
4 2  32  7
52  4 2  9
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4.
5.
6.
Math 20
Reasoning and Geometry Review
Decide whether each conjecture is true or false. If it is true, explain why. If it is
false, prove a counterexample.
a.
The sum of two prime numbers is an even number.
b.
Odd whole numbers less than 10 are prime.
c.
Multiples of 4 are divisible by 8.
d.
If a, b, c, and d are any positive real numbers such that a  b and c  d , then
ac  bd .
Use deductive reasoning. Complete each conclusion.
a.
all cats have four paws. Fluffy is a cat. Therefore….
b.
all months have more than 27 days. February is a month. Therefore….
c.
all the students in Susan’s class have at least one pet at home. Julie is in
Susan’s class. Therefore….
d.
all quadrilaterals have four vertices. A parallelogram is a quadrilateral.
Therefore ….
a.
Prove that the difference between the squares of two odd numbers is always
divisible by 4.
b.
Prove that the sum of any two odd numbers is an even number.
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6.
7.
Math 20
c.
Choose a number. Triple the number. Add 2. Add the original number.
Add 2. Divide by 4. Subtract 1. Repeat this sequence of steps. Make a
conjecture about the final number. Then prove your conjecture using
deductive reasoning.
d.
Prove that the product of any two odd numbers is an odd number.
On a number line, indicate the location of the numbers corresponding to each
statement.
a.
8.
9.
Reasoning and Geometry Review
x4
b.
x4
Which numbers x satisfy this statement?
a.
x is a multiple of 2 and x is a factor of 54.
b.
x is a factor of 35 or x is a factor of 56.
On a number line, indicate the location of the numbers described by each statement.
a.
x  1 and x  7
b.
x  1 and x  7
c.
x  1 or x  7
d.
x  1 or x  7
e.
x  5 and x  4
f.
x  1 or x  3
g.
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h.
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10.
Math 20
a.
Reasoning and Geometry Review
A census at a small school revealed these data. The total enrolment at the
school was 121 students.
37 students were in the drama club
40 students were on the debating team
56 students were on a sports team
13 students were on a sports team and were in the drama club
16 students were on a sports team and were on the debating team
12 students were in the drama club and were on the debating team
8 students were in all three activities
i.
How many students participated in just one activity?
ii.
How many students participated in just two activities?
iii.
How many students participated in none of the activities?
iv.
How many students were in the drama club and on a sports team, but not on
the debating team?
b.
Students were asked about the language course they took in high school. The
results of the survey and the total enrollment in each course were as follows.
Total enrollment in French: 131 students
Total enrollment in Spanish: 115 students
Total enrollment in Italian: 82 students
i.
ii.
Draw a Venn diagram to summarize this data.
The survey did not ask students if they were taking two of the three language
courses. Determine these three unknown values.
iii.
How many students were surveyed?
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11.
Math 20
a.
Of the students in grade 11 at a certain high school, 76 are enrolled in
physical education, 24 are enrolled in music, and 10 are enrolled in both
physical education and music. If there are 15 students in grade 11 who are
not enrolled in physical education or music, how many students are in grade
11?
b.
Twenty-six people are surveyed at a mall. All of the people shopped at the
hardware store or the dollar store. Sixteen people shopped at the hardware
store and 20 shopped at the dollar store. How many people shopped:
i.
ii.
iii.
12.
13.
Reasoning and Geometry Review
at both stores?
only at the hardware store?
only at the dollar store?
Write each statement in an “If…then” form. Is the true or false? Explain.
a.
A rhombus is a square.
b.
The squares of all odd numbers are odd.
Consider the statement “The lowest common multiple of two prime numbers is their
product.”
a.
Write the statement in an “If…then” form. Is it true or false? Explain.
b.
Write the converse and the contrapositive of the statement. Is each new
statement true or false? Explain.
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14.
Math 20
Reasoning and Geometry Review
Consider the statement “A trapezoid has exactly one pair of parallel sides.”
a.
Write the statement in an “If…then” form. Is it true or false? Explain.
b.
Write the converse and the contrapositive of the statement. Is each new
statement true or false? Explain.
15.
In the diagram, BC AD and BC  AD, Prove that ABC  CDA
16.
In the diagram, BC  BE and BCA  BED. Prove that AC  DE.
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Math 20
Reasoning and Geometry Review
17.
In the diagram, D is any point on the angle bisector of ABC . Line segment
DA  AB and DC  BC. Prove that DAB  DCB.
18.
Prove the diagonals of a parallelogram bisect each other.
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Math 20
Reasoning and Geometry Review
ANSWER KEY
1.
The exterior angle of a triangle is equal to the sum of the interior opposite angles.
2.
The angles opposite the equal sides in an isosceles triangle are equal.
3.
The difference in the squares of two consecutive whole numbers is equal to the sum
of the whole numbers.
4.
a.
The conjecture is false. Since 2 and 3 are both prime numbers and their
sum is 5 which is an odd number.
b.
The conjecture is false. One is an odd number which is not prime.
c.
d.
The conjecture is false. 12 is a multiple of 4 but not divisible by 8
The conjecture is true.
5.
a.
c.
Fluffy has four paws
Julie has at least one pet at home
6.
a.
 2n  1   2m  1
2
b.
d.
February has more than 27 days.
A parallelogram has 4 vertices
2
4n 2  4n  1  4m 2  4m  1

4 n2  n  m2  m

Since 4 is a factor of the product, the product is divisible by 4.
b.
2n  1  2m  1
2m  2n  2
2  m  n  1
Since 2 is a factor of the product, the sum is an even number.
c.
d.
a  3a  3a  2  3a  2  a  4a  4
 a 1  a
 2m  1 2n  1
4mn  2m  2n  1
2  2mn  m  n   1
Since the answer is one more than twice a number, it must be an odd number
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7.
Math 20
Reasoning and Geometry Review
a.
b.
8.
a.
9.
a.
10.
b.
2, 6, 18, 54
1, 2, 4,5, 7,8,14, 28,56
b.
c.
The combined statement does not describe any numbers on the number line
d.
e.
f.
g.
h.
a.
b.
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i.
75
ii.
17
iii.
46
iv.
5
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b.
Math 20
Reasoning and Geometry Review
Let x be the number of students enrolled in both French and Spanish
Let y be the number of students enrolled in both French and Italian
Let z be the number of students enrolled in both Spanish and Italian
French enrollment = x  y  41  56  131
Spanish enrollment = x  z  20  56  115
Italian enrollment = y  z  3  56  82
x  y  34, x  z  39, y  z  23
So:
And: y  z  5, y  z  23
 x  25, y  9, z  14
11.
12.
a.
b.
105
i.
a.
If a figure is a rhombus, then it is a square. False. A square is a special
rhombus, one with right angles.
b.
10
ii.
6
iii.
10
If a number is odd, then its square is odd. True,  2m  1  4m 2  4m  1.
2
Because 2 is not a factor of the product, the product is always odd.
13.
a.
If two numbers are prime, then their lowest common multiple is their product.
True. Each of the numbers has no factors other than 1 and itself. Therefore,
they have no common factors other than 1 and no common multiple that is
less than their product.
b.
Converse: If the lowest common multiple of two numbers is their product,
then the numbers are prime. False. Consider this counterexample: the
lowest common multiple of 4 and 9 is 36, but neither number is a prime.
Contrapositive: If the lowest common multiple of two numbers is not their
product, then the numbers are not prime. True. Since the lowest common
multiple is not their product, the two numbers must have at least one common
factor other than 1. This means at least one of the number is not prime. We
also know the contrapositive is true because the original statement was true.
14.
a.
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If a figure is a trapezoid, then it has exactly one pair of parallel sides. True.
If it had more than one pair of parallel sides, it would be a parallelogram, and
if it had no parallel sides, then it would not be a trapezoid.
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14.
b
Math 20
Reasoning and Geometry Review
Converse: If a figure has exactly one pair of parallel sides, then it is a
trapezoid. False. The figure may not even be a quadrilateral.
Contrapositive: If a figure does not have exactly one pair of parallel sides,
then it is not a trapezoid. True. The reasoning is the same as for the
statement in part a.
15.
By the Parallel Lines Theorem, BCA  DAC .
In ABC and CDA, AC  AC , BCA  DAC , and BC  DA
16.
ABC  CDA  SAS 
By the Opposite Angles Theorem, ABC  DBE.
In ABC and DBE, ABC  DBE, BC  BE,
BCA  BED. ABC  DBE.  ASA
Since the triangles are congruent, AC  DE.
17.
Answer may vary
DAB  DCB
given
DBA  DBC
given
BD  BD
common side
DAB  DCB
18.
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 AAS 