Download Geo - CH2 Practice Test

Geo - CH2 Practice Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
1. Find the next item in the pattern 2, 3, 5, 7, 11, ...
a. 13
c. 15
b. 12
d. 17
2. The table shows the population 65 years and over by age and sex according to the US Census
Bureau, Census 2000 Summary file. Make a conjecture based on the data.
Population 65 Years and Over by Age and Sex: 2000
(numbers in thousands)
65 to 74 years
75 to 84 years
85 years and over
Women
10,088
7,482
3,013
Men
8,303
4,879
1,227
____
a. Women outnumbered men in the 65 years and over population.
b. Men outnumbered women in the 65 years and over population.
c. There are more 65 years old and over in 2000 than in previous years.
d. There are fewer 65 years old and over in 2000 than in previous years.
3. Show that the conjecture is false by finding a counterexample.
If a > b, then a > 0.
b
a. a = 11, b = −3
b. a = 11, b = 3
____
____
c. a = 3, b = 11
d. a = −11, b = 3
4. There is a myth that a duck’s quack does not echo. A group of scientists observed a duck in a
special room, and they found that the quack does echo. Therefore, the myth is false.
Is the conclusion a result of inductive or deductive reasoning?
a. Since the conclusion is based on a pattern of observation, it is a result of inductive
reasoning.
b. Since the conclusion is based on a pattern of observation, it is a result of deductive
reasoning.
c. Since the conclusion is based on logical reasoning from scientific research, it is a
result of inductive reasoning.
d. Since the conclusion is based on logical reasoning from scientific research, it is a
result of deductive reasoning.
5. Determine if the conjecture is valid by the Law of Syllogism.
Given: If you are in California, then you are in the west coast. If you are in Los Angeles, then you
are in California.
Conjecture: If you are in Los Angeles, then you are in the west coast.
a. No, the conjecture is not valid.
b. Yes, the conjecture is valid.
____
6. A gardener has 26 feet of fencing for a garden. To find the width of the rectangular garden, the
gardener uses the formula P = 2l + 2w, where P is the perimeter, l is the length, and w is the width
of the rectangle. The gardener wants to fence a garden that is 8 feet long. How wide is the garden?
Solve the equation for w, and justify each step.
P = 2l + 2w
26 = 2(8) + 2w
26 = 16 + 2w
−16 = −16
10 = 2w
10 = 2w
2
2
5=w
w=5
____
Given equation
[1]
Simplify.
Subtraction Property of Equality
Simplify.
[2]
Simplify.
Symmetric Property of Equality
a. [1] Substitution Property of Equality
[2] Division Property of Equality
The garden is 5 ft wide.
b. [1] Simplify
[2] Division Property of Equality
The garden is 5 ft wide.
7. Write a justification for each step.
c. [1] Substitution Property of Equality
[2] Subtraction Property of Equality
The garden is 5 ft wide.
d. [1] Subtraction Property of Equality
[2] Simplify
The garden is 5 ft wide.
m∠JKL = 100°
m∠JKL = m∠JKM + m∠MKL
100° = (6x + 8)° + (2x − 4)°
100 = 8x + 4
96 = 8x
12 = x
x = 12
[1]
Substitution Property of Equality
Simplify.
Subtraction Property of Equality
[2]
Symmetric Property of Equality
a. [1] Transitive Property of Equality
[2] Division Property of Equality
b. [1] Angle Addition Postulate
[2] Division Property of Equality
c. [1] Angle Addition Postulate
[2] Simplify.
d. [1] Segment Addition Postulate
[2] Multiplication Property of Equality
____
8. Fill in the blanks to complete the two-column proof.
Given: ∠1 and ∠2 are supplementary. m∠1 = 135°
Prove: m∠2 = 45°
Proof:
Statements
1. ∠1 and ∠2 are supplementary.
2. [1]
3. m∠1 + m∠2 = 180°
4. 135° + m∠2 = 180°
5. m∠2 = 45°
Reasons
1. Given
2. Given
3. [2]
4. Substitution Property
5. [3]
a. [1] m∠2 = 135°
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
b. [1] m∠1 = 135°
[2] Definition of supplementary angles
[3] Substitution Property
c. [1] m∠1 = 135°
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
d. [1] m∠1 = 135°
[2] Definition of complementary angles
[3] Subtraction Property of Equality
____
____
2
2
9. Two angles with measures (2x + 3x − 5)° and (x + 11x − 7)° are supplementary. Find the value of
x and the measure of each angle.
a. x = 5; 60°; 30°
c. x = 5; 60°; 120°
b. x = 6; 85°; 95°
d. x = 4; 40°; 90°
10. Two lines intersect to form two pairs of vertical angles. ∠1 with measure (20x + 7)º and ∠3 with
measure (5x + 7y + 49)º are vertical angles. ∠2 with measure (3x − 2y + 30)º and ∠4 are vertical
angles. Find the values x and y and the measures of all four angles.
a. x = 6; y = 10; 127°; 127°; 28°; 28°
c. x = 5; y = 5; 107°; 107°; 73°; 73°
b. x = 8; y = 11, 167°; 167°; 13°; 13°
d. x = 7; y = 9; 147°; 147°; 33°; 33°
Numeric Response
11. Find a value for x that provides a counterexample for this conjecture.
For all real numbers x, 4x − 8 = 4 .
3x − 6
3
12. In the quadrilateral, m∠1 + m∠2 + m∠3 + m∠4 = 360°. If m∠2 = 3m∠1, m∠3 = m∠1 + 6, and
m∠4 = m∠1, find m∠3 in degrees.
Matching
Match each vocabulary term with its definition.
a. conjecture
b. inductive reasoning
c. deductive reasoning
d. conclusion
e. biconditional statement
f. hypothesis
g. counterexample
h. conditional statement
____
____
____
____
____
13.
14.
15.
16.
17.
an example that proves that a conjecture or statement is false
a statement that is believed to be true
the part of a conditional statement following the word then
the part of a conditional statement following the word if
the process of reasoning that a rule or statement is true because specific cases are true
Match each vocabulary term with its definition.
a. conclusion
b. converse
c. inverse
d. negation
e. hypothesis
f. truth value
g. contrapositive
____
____
18. a statement can have a truth value of true (T) or false (F)
19. operations that undo each other
____
____
____
20. the contradiction of statement by using “not,” written as ∼
21. the statement formed by exchanging the hypothesis and conclusion of a conditional statement
22. the statement formed by both exchanging and negating the hypothesis and conclusion
Geo - CH2 Practice Test
Answer Section
MULTIPLE CHOICE
1. ANS: A
The prime numbers make up the pattern. The next prime is 13.
Feedback
A
B
C
D
Correct!
The prime numbers make up the pattern. What is the next prime number?
The prime numbers make up the pattern. What is the next prime number?
Is there a prime number that is greater than 11 but smaller than 17?
PTS: 1
DIF: Basic
REF: Page 74
OBJ: 2-1.1 Identifying a Pattern
NAT: 12.5.1.a
TOP: 2-1 Using Inductive Reasoning to Make Conjectures
2. ANS: A
For every age group 65 years and over, the number of women is greater than the number of men.
The data supports the conjecture that women outnumbered men in the 65 years and over
population.
Feedback
A
B
C
D
Correct!
Look at the table's data. Are there more men or women 65 years and over?
Look at the table's data. Is there any information about previous years?
Look at the table's data. Is there any information about previous years?
PTS: 1
DIF: Average
REF: Page 75
OBJ: 2-1.3 Application
NAT: 12.3.5.a
TOP: 2-1 Using Inductive Reasoning to Make Conjectures
3. ANS: A
Pick values for a and b that follow the condition a > b. Then substitute them into the second
inequality to see if the conjecture holds.
Values of a and b
a>b
Let a = 4 and b = 1.
4>1
Let a = 11 and b = 3.
11 > 3
Let a = 11 and b = −3.
11 > −3
a >0
b
4 >0
1
11 > 0
3
11 < 0
−3
Conclusion
The conjecture holds.
The conjecture holds.
The conjecture is false.
a = 11 and b = −3 is a counterexample.
The conjecture is false when a is positive and b is negative.
Feedback
A
B
C
D
Correct!
In this case, a/b is greater than zero, so it is not a counterexample.
In this case, a is not greater than b. The counterexample should have a > b and
a/b less than or equal to 0.
In this case, a is not greater than b. a > b is the condition of the conjecture. The
counterexample should have a > b and a/b less than or equal to 0.
PTS: 1
DIF: Average
REF: Page 76
OBJ: 2-1.4 Finding a Counterexample
NAT: 12.3.5.a
TOP: 2-1 Using Inductive Reasoning to Make Conjectures
4. ANS: A
The scientists determined the myth was false because they heard an echo by observing the duck.
Inductive reasoning is based on a pattern of observation.
Feedback
A
B
C
D
Correct!
Deductive reasoning is based on logical reasoning.
Inductive reasoning is based on observation.
This conclusion was based on observation instead of logical reasoning.
PTS: 1
DIF: Basic
REF: Page 88
OBJ: 2-3.1 Application
NAT: 12.3.5.a
TOP: 2-3 Using Deductive Reasoning to Verify Conjectures
5. ANS: B
Let p, q, and r represent the following.
p: You are in California.
q: You are in the west coast.
r: You are in Los Angeles.
You are given that p → q and r → p
Since p is the conclusion of the second statement and the hypothesis of the first statement, reorder
the statements like this r → p and p → q.
By the Law of Syllogism, if r → p and p → q are true, then r → q is true.
r → q is the statement, If you are in Los Angeles, then you are in the west coast.
Feedback
A
B
The Law of Syllogism states that if (if p, then q) and (if q, then r) are true
statements, then (if p, then r) is a true statement.
Correct!
PTS: 1
DIF: Average
REF: Page 89
OBJ: 2-3.3 Verifying Conjectures by Using the Law of Syllogism
NAT: 12.3.5.a
6. ANS: A
P = 2l + 2w
26 = 2(8) + 2w
26 = 16 + 2w
−16 = −16
10 = 2w
10 = 2w
2
2
5=w
w=5
TOP: 2-3 Using Deductive Reasoning to Verify Conjectures
Given equation
[1] Substitution Property of Equality
Simplify.
Subtraction Property of Equality
Simplify.
[2] Division Property of Equality
Simplify.
Symmetric Property of Equality
Feedback
A
B
C
D
Correct!
The variables P and l are substituted, not simplified. Use the Substitution
Property.
Check the properties.
Check the justifications.
PTS: 1
DIF: Average
REF: Page 105
OBJ: 2-5.2 Problem-Solving Application NAT: 12.5.2.e
TOP: 2-5 Algebraic Proof
7. ANS: B
m∠JKL = m∠JKM + m∠MKL
[1] Angle Addition Postulate
100° = (6x + 8)° + (2x − 4)°
Substitution Property of Equality
Simplify.
100 = 8x + 4
Subtraction Property of Equality
96 = 8x
[2] Division Property of Equality
12 = x
Symmetric Property of Equality
x = 12
Feedback
A
B
C
D
Check the properties.
Correct!
Check the justifications.
The Segment Addition Postulate refers to segments, not angles.
PTS:
OBJ:
TOP:
8. ANS:
Proof:
1
DIF: Average
REF: Page 106
2-5.3 Solving an Equation in Geometry
2-5 Algebraic Proof
C
Statements
1. ∠1 and ∠2 are supplementary.
NAT: 12.5.2.e
Reasons
1. Given
2. m∠1 = 135°
3. m∠1 + m∠2 = 180°
4. 135° + m∠2 = 180°
5. m∠2 = 45°
2. Given
3. Definition of supplementary angles
4. Substitution Property
5. Subtraction Property of Equality
Feedback
A
B
C
D
Check to the given information.
To get from Step 4 to Step 5, use subtraction, not substitution.
Correct!
The angles are supplementary, not complementary.
PTS: 1
DIF: Average
REF: Page 111
OBJ: 2-6.2 Completing a Two-Column Proof
TOP: 2-6 Geometric Proof
9. ANS: B
Step 1 Create an equation
The angles are supplements and their sum equals 180°.
NAT: 12.3.5.a
(2x2 + 3x − 5) + (x2 + 11x − 7) = 180
Step 2 Solve the equation
3x2 + 14x − 12 = 180
3x2 + 14x − 192 = 0
(3x + 32)(x − 6) = 0
x = − 323 or 6.
When x = − 323 , the measurement of the second angle is
x2 + 11x − 7 = −10.6°.
Angles cannot have negative measurements, so x = 6.
Step 3 Solve for the required values
2
2
The measurement of the first angle is 2x + 3x − 5 = 2(6) + 3(6) − 5 = 85°.
2
2
The measurement of the second angle is x + 11x − 7 = (6) + 11(6) − 7= 95°.
Feedback
A
B
C
D
The angles are supplements. Use the definition of supplements to solve for x.
Correct!
Check for algebra mistakes. When x equals 5, the second angle is not 120
degrees.
The angles are supplements. Use the definition of supplements to solve for x.
PTS: 1
DIF: Advanced
10. ANS: D
Step 1 Create a system of equations.
NAT: 12.2.1.f
TOP: 2-6 Geometric Proof
m∠1 = m∠3
20x + 7 = 5x + 7y + 49
15x − 7y = 42
The sum of the measures of supplementary angles equals 180°.
m∠1 + ∠2 = 180
20x + 7 + 3x − 2y + 30 = 180
23x − 2y = 143
Create a system of equations.
15x − 7y = 42
23x − 2y = 143
Step 2 Solve the system of equations.
15x − 7y = 42
23x − 2y = 143
−30x + 14y = −84
161x − 14y = 1001
131x = 917
x=7
Multiply the first equation by −2.
Multiply the second equation by 7.
Add the two equations together.
Divide both sides by 131.
Solve for y.
Substitute x = 7 into 15x − 7y = 42.
15(7) − 7y = 42
y=9
The values are x = 7 and y = 9.
Step 3 Solve for the four angles.
Angle 1: (20(7) + 7)° = 147°
Angle 2: (3(7) − 2(9) + 30)° = 33°
Angle 3: (5(7) + 7(9) + 49)° = 147°
Angle 4 and angle 2 are vertical and thus have equal measures.
The measurement of angle 4 is 33°.
The measures of all four angles are 147°, 147°, 33°, and 33°.
Feedback
A
B
C
D
Use the definitions of supplementary and vertical angles to create a solvable
system of equations.
Use the definitions of supplementary and vertical angles to create a solvable
system of equations.
Use the definitions of supplementary and vertical angles to create a solvable
system of equations.
Correct!
PTS: 1
DIF: Advanced
NAT: 12.2.1.f
TOP: 2-7 Flowchart and Paragraph Proofs
NUMERIC RESPONSE
11. ANS: 2
PTS: 1
DIF: Advanced
NAT: 12.3.5.a
TOP: 2-1 Using Inductive Reasoning to Make Conjectures
12. ANS: 65
PTS:
1
DIF:
Average
NAT: 12.2.1.f
TOP: 2-5 Algebraic Proof
MATCHING
13. ANS:
TOP:
14. ANS:
TOP:
15. ANS:
TOP:
16. ANS:
TOP:
17. ANS:
TOP:
G
PTS: 1
DIF: Basic
2-1 Using Inductive Reasoning to Make Conjectures
A
PTS: 1
DIF: Basic
2-1 Using Inductive Reasoning to Make Conjectures
D
PTS: 1
DIF: Basic
2-2 Conditional Statements
F
PTS: 1
DIF: Basic
2-2 Conditional Statements
B
PTS: 1
DIF: Basic
2-1 Using Inductive Reasoning to Make Conjectures
REF: Page 75
18. ANS:
TOP:
19. ANS:
TOP:
20. ANS:
TOP:
21. ANS:
TOP:
22. ANS:
TOP:
F
PTS: 1
2-2 Conditional Statements
C
PTS: 1
2-2 Conditional Statements
D
PTS: 1
2-2 Conditional Statements
B
PTS: 1
2-2 Conditional Statements
G
PTS: 1
2-2 Conditional Statements
REF: Page 74
REF: Page 81
REF: Page 81
REF: Page 74
DIF:
Basic
REF: Page 82
DIF:
Basic
REF: Page 83
DIF:
Basic
REF: Page 82
DIF:
Basic
REF: Page 83
DIF:
Basic
REF: Page 83