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Name: ________________________ Class: ___________________ Date: __________ ID: A

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Chapter 2 review 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. Complete the conjecture.

The sum of two odd numbers is _____.

a. even c. sometimes odd, sometimes even

b. odd d. even most of the time

____ 3. 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.

____ 4. Show that the conjecture is false by finding a counterexample.

If a > b, then a

b> 0.

a. a = 11, b = −3 c. a = 3, b = 11

b. a = 11, b = 3 d. a = −11, b = 3

____ 5. Make a table of values for the rule x2

− 16x + 64 when x is an integer from 1 to 6. Make a conjecture about

the type of number generated by the rule. Continue your table. What value of x generates a counterexample?

a. The pattern appears to be an decreasing set of perfect squares.

x = 9 generates a counterexample.

b. The pattern appears to be a decreasing set of prime numbers.


c. The pattern appears to be a decreasing set of perfect squares.


d. The pattern appears to be an increasing set of perfect squares.


Name: ________________________ ID: A

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____ 6. Identify the hypothesis and conclusion of the conditional statement.

If it is raining then it is cloudy.

a. Hypothesis: It is raining.

Conclusion: It is cloudy.

b. Hypothesis: It is cloudy.

Conclusion: It is raining.

c. Hypothesis: Clouds make rain.

Conclusion: Rain does not make clouds.

d. Hypothesis: Rain and clouds happen together.

Conclusion: Rain and clouds do not happen together..

____ 7. Write a conditional statement from the statement.

A horse has 4 legs.

a. If it has 4 legs then it is a horse. c. If it is a horse then it has 4 legs.

b. Every horse has 4 legs. d. It has 4 legs and it is a horse.

____ 8. Determine if the conditional statement is true. If false, give a counterexample. If a figure has four sides, then

it is a square.

a. True.

b. False; A rectangle has four sides, and it is not a square.

____ 9. Write the converse, inverse, and contrapositive of the conditional statement.

If an animal is a bird, then it has two eyes.

a. Converse: If an animal is not a bird, then it does not have two eyes.

Inverse: If an animal does not have two eyes, then it is not a bird.

Contrapositive: If an animal is a bird, then it has two eyes.

b. Converse: If an animal has two eyes, then it is a bird.

Inverse: If an animal is not a bird, then it does not have two eyes.

Contrapositive: If an animal does not have two eyes, then it is not a bird.

c. Converse: If an animal does not have two eyes, then it is not a bird.

Inverse: If an animal is not a bird, then it does not have two eyes.

Contrapositive: If an animal has two eyes, then it is a bird.

d. Converse: All birds have two eyes.

Inverse: All animals have two eyes.

Contrapositive: All birds are animals, and animals have two eyes.

____ 10. How many true conditional statements may be written using the following statements?

n is a rational number.

n is an integer.

n is a whole number.

a. 2 conditional statements c. 4 conditional statements

b. 3 conditional statements d. 5 conditional statements

Name: ________________________ ID: A

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____ 11. 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.

____ 12. Determine if the conjecture is valid by the Law of Detachment.

Given: If Tommy makes cookies tonight, then Tommy must have an oven. Tommy has an oven.

Conjecture: Tommy made cookies tonight.

a. The conjecture is valid, because if Tommy didn’t have an oven then he didn’t make

cookies tonight

b. The conjecture is not valid, because if Tommy didn’t have an oven then he didn’t make

cookies tonight.

c. The conjecture is valid, because Tommy could have an oven but he could make

something besides cookies tonight.

d. The conjecture is not valid, because Tommy could have an oven but he could make

something besides cookies tonight.

____ 13. 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.

____ 14. Draw a conclusion from the given information.

Given: If two lines are perpendicular, then they form right angles. If two lines meet at a 90° angle, then they

are perpendicular. Two lines meet at a 90° angle.

a. Conclusion: The lines are perpendicular.

b. Conclusion: The lines are perpendicular, meet at a 90° angle, and form a right angle.

c. Conclusion: The lines meet at a 90° angle.

d. Conclusion: The lines form a right angle.

Name: ________________________ ID: A

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____ 15. Consider the two conditional statements. Draw a conclusion from the given conditional statements, and write

the contrapositive of each conditional statement. Then, draw a conclusion from the two contrapositives. How

does the first conclusion relate to the second conclusion?

If you are eating a banana, then you are eating fruit.

If you are eating fruit, then you are eating food.

a. If you are eating a banana, then you are eating food.

If you are not eating a banana, then you are not eating fruit. If you are not eating fruit,

then you are not eating food.

If you are not eating a banana, then you are not eating food.

The second conclusion is the contrapositive of the first conclusion.

b. If you are eating a banana, then you are eating food.

If you are eating fruit, then you are eating a banana. If you are eating food, then you are

eating fruit.

If you are eating food, then you are eating a banana.


c. If you are eating a banana, then you are eating food.

If you are not eating fruit, then you are not eating a banana. If you are not eating food,

then you are not eating fruit.

If you are not eating food, then you are not eating a banana.


d. If you are eating food, then you are eating a banana.

If you are not eating fruit, then you are not eating a banana. If you are not eating food,

then you are not eating fruit.

If you are not eating food, then you are not eating a banana.


____ 16. Write the conditional statement and converse within the biconditional.

A rectangle is a square if and only if all four sides of the rectangle are equal length.

a. Conditional: If all four sides of the rectangle are equal length, then it is a square.

Converse: If a rectangle is a square, then its four sides are equal length.

b. Conditional: If a rectangle is a square, then it is also a rhombus.

Converse: If a rectangle is a rhombus, then it is also a square.

c. Conditional: If all four sides are equal length, then all four angles are 90°.

Converse: If all four angles are 90°, then all four sides are equal length.

d. Conditional: If a rectangle is not a square, then its sides are of different lengths.

Converse: If the sides are of different lengths, then the rectangle is not a square.

Name: ________________________ ID: A

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____ 17. For the conditional statement, write the converse and a biconditional statement.

If a figure is a right triangle with sides a, b, and c, then a2

+ b2

= c2.

a. Converse: If a figure is not a right triangle with sides a, b, and c, then a2

+ b2

≠ c2.

Biconditional: A figure is a right triangle with sides a, b, and c if and only if

a2

+ b2

= c2.

b. Converse: If a2

+ b2

= c2, then the figure is a right triangle with sides a, b, and c.


a2

+ b2

= c2.

c. Converse: If a2

+ b2

≠ c2, then the figure is not a right triangle with sides a, b, and c.

Biconditional: A figure is not a right triangle with sides a, b, and c if and only if

a2

+ b2

≠ c2

d. Converse: If a2

+ b2

≠ c2, then the figure is not a right triangle with sides a, b, and c.


a2

+ b2

= c2.

____ 18. Determine if the biconditional is true. If false, give a counterexample.

A figure is a square if and only if it is a rectangle.

a. The biconditional is true.

b. The biconditional is false. A rectangle does not necessarily have four congruent sides.

c. The biconditional is false. All squares are parallelograms with four 90° angles.

d. The biconditional is false. A rectangle does not necessarily have four 90° angles.

____ 19. Write the definition as a biconditional.

An acute angle is an angle whose measure is less than 90°.

a. An angle is acute if its measure is less than 90°.

b. An angle is acute if and only if its measure is less than 90°.

c. An angle’s measure is less than 90° if it is acute.

d. An angle is acute if and only if it is not obtuse.

____ 20. What is the truth value of the biconditional formed from the conditional, “If B is the midpoint of A and C,

then AB = BC.” Explain.

a. The conditional is true.

The converse, “If AB = BC then B is the midpoint of AC” is false.

Since the conditional is true but the converse is false, the biconditional is false.

b. The conditional is true.

The converse, “If AB = BC then B is the midpoint of AC” is true.

Since the conditional is true and the converse is true, the biconditional is true.

c. The conditional is false.

The converse, “If AB = BC then B is the midpoint of AC” is false.

Since the conditional is false and the converse is false, the biconditional is true.

d. The conditional is false.

The converse, “If AB = BC then B is the midpoint of AC” is true.

Since the conditional is false and the converse is true, the biconditional is false.

Name: ________________________ ID: A

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____ 21. Solve the equation 4x − 6 = 34. Write a justification for each step.

4x − 6 = 34 Given equation

+6 +6 [1]

4x = 40 Simplify.

4x

4=

40

4[2]

x = 10 Simplify.

a. [1] Substitution Property of Equality;

[2] Division Property of Equality

c. [1] Division Property of Equality;

[2] Subtraction Property of Equality

b. [1] Addition Property of Equality;


d. [1] Addition Property of Equality;

[2] Reflexive Property of Equality

____ 22. 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 Given equation

26 = 2(8) + 2w [1]

26 = 16 + 2w

−16 = −16

10 = 2w

Simplify.

Subtraction Property of Equality

Simplify.

10

2=

2w

2[2]

5 = w Simplify.

w = 5 Symmetric Property of Equality

a. [1] Substitution Property of Equality


The garden is 5 ft wide.

c. [1] Substitution Property of Equality



b. [1] Simplify



d. [1] Subtraction Property of Equality

[2] Simplify


Name: ________________________ ID: A

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____ 23. Write a justification for each step.

m∠JKL = 100°

m∠JKL = m∠JKM + m∠MKL [1]

100° = (6x + 8)° + (2x − 4)° Substitution Property of Equality

100 = 8x + 4 Simplify.

96 = 8x Subtraction Property of Equality

12 = x [2]

x = 12 Symmetric Property of Equality

a. [1] Transitive Property of Equality


b. [1] Angle Addition Postulate


c. [1] Angle Addition Postulate

[2] Simplify.

d. [1] Segment Addition Postulate

[2] Multiplication Property of Equality

____ 24. Identify the property that justifies the statement.

AB ≅ CD and CD ≅ EF. So AB ≅ EF.

a. Reflexive Property of Congruence c. Symmetric Property of Congruence

b. Substitution Property of Equality d. Transitive Property of Congruence

Name: ________________________ ID: A

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____ 25. Write a justification for each step, given that EG = FH .

EG = FH Given information

EG = EF + FG [1]

FH = FG + GH Segment Addition Postulate

EF + FG = FG + GH [2]

EF = GH Subtraction Property of Equality

a. [1] Angle Addition Postulate


b. [1] Substitution Property of Equality

[2] Transitive Property of Equality

c. [1] Segment Addition Postulate

[2] Definition of congruent segments

d. [1] Segment Addition Postulate

[2] Substitution Property of Equality

Name: ________________________ ID: A

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____ 26. 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 Reasons

1. ∠1 and ∠2 are supplementary. 1. Given

2. [1] 2. Given

3. m∠1 + m∠2 = 180° 3. [2]

4. 135° + m∠2 = 180° 4. Substitution Property

5. m∠2 = 45° 5. [3]

a. [1] m∠2 = 135°

[2] Definition of supplementary angles


b. [1] m∠1 = 135°


[3] Substitution Property

c. [1] m∠1 = 135°



d. [1] m∠1 = 135°

[2] Definition of complementary angles


Name: ________________________ ID: A

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____ 27. Use the given plan to write a two-column proof.

Given: m∠1 + m∠2 = 90°, m∠3 + m∠4 = 90°, m∠2 = m∠3

Prove: m∠1 = m∠4

Plan: Since both pairs of angle measures add to 90°, use substitution to show that the sums of both pairs are

equal. Since m∠2 = m∠3, use substitution again to show that sums of the other pairs are equal. Use the

Subtraction Property of Equality to conclude that m∠1 = m∠4.

Complete the proof.

Proof:

Statements Reasons

1. m∠1 + m∠2 = 90° 1. Given

2. [1] 2. Given

3. m∠1 + m∠2 = m∠3 + m∠4 3. Substitution Property

4. m∠2 = m∠3 4. Given

5. m∠1 + m∠2 = m∠2 + m∠4 5. [2]

6. m∠1 = m∠4 6. [3]

a. [1] m∠3 + m∠4 = 90°



b. [1] m∠5 + m∠6 = 90°



c. [1] m∠3 + m∠4 = 90°



d. [1] m∠5 + m∠6 = 90°

[2] Addition Property of Equality


____ 28. Two angles with measures (2x2

+ 3x − 5)° and (x2

+ 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°

Name: ________________________ ID: A

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____ 29. Use the given flowchart proof to write a two-column proof of the statement AF ≅ FD.

Flowchart proof:

AB = CD;BF = FC

AB + BF = AF

FC + CD = FD

Given

Segment

Addition

Postulate

AB + BF =

FC + CDAF = FD AF ≅ FD

Addition

Property of

Equality

Substitution Definition of

congruent segments

Complete the proof.

Two-column proof:

Statements Reasons

1. AB = CD; BF = FC 1. Given

2. [1] 2. Addition Property of Equality

3. [2] 3. Segment Addition Postulate

4. AF = FD 4. Substitution

5. AF ≅ FD 5. Definition of congruent segments

a. [1] AB + BF = AF ; FC + CD = FD

[2] AF = FD

b. [1] AF = FD

[2] AB + BF = FC + CD

c. [1] AB = CD; BF = FC

[2] AB + BF = FC + CD

d. [1] AB + BF = FC + CD

[2] AB + BF = AF ;FC + CD = FD

Name: ________________________ ID: A

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____ 30. Use the given two-column proof to write a flowchart proof.

Given: ∠1 ≅ ∠4

Prove: m∠2 = m∠3

Two-column proof:

Statements Reasons

1. ∠1 ≅ ∠4 1. Given

2. ∠1 and ∠2 are supplementary. ∠3 and ∠4

are supplementary.

2. Definition of linear pair

3. ∠2 ≅ ∠3 3. Congruent Supplements Theorem

4. m∠2 = m∠3 4. Definition of congruent segments

Complete the proof.

Flowchart proof:

∠1 ≅ ∠4

Given

[1] ∠2 ≅ ∠3 m∠2 = m∠3

Definition of linear pair [2] Definition of

congruent segments

a. [1] ∠1 and ∠2 are supplements; ∠3 and ∠4 are supplementary

[2] Congruent Complements Theorem

b. [1] ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary

[2] Congruent Supplements Theorem

c. [1] ∠2 ≅ ∠3

[2] Definition of congruent segments

d. [1] Definition of congruent segments

[2] Congruent Supplements Theorem

Name: ________________________ ID: A

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____ 31. Use the given paragraph proof to write a two-column proof.

Given: ∠BAC is a right angle. ∠1 ≅ ∠3

Prove: ∠2 and ∠3 are complementary.

Paragraph proof:

Since ∠BAC is a right angle, m∠BAC = 90° by the definition of a right angle. By the Angle Addition

Postulate, m∠BAC = m∠1 + m∠2. By substitution, m∠1 + m∠2 = 90°. Since ∠1 ≅ ∠3, m∠1 = m∠3 by the

definition of congruent angles. Using substitution, m∠3 + m∠2 = 90°. Thus, by the definition of

complementary angles, ∠2 and ∠3 are complementary.

Complete the proof.

Two-column proof:

Statements Reasons

1. ∠BAC is a right angle. ∠1 ≅ ∠3 1. Given

2. m∠BAC = 90° 2. Definition of a right angle

3. m∠BAC = m∠1 + m∠2 3. [1]

4. m∠1 + m∠2 = 90° 4. Substitution

5. m∠1 = m∠3 5. [2]


7. ∠2 and ∠3 are complementary. 7. Definition of complementary angles

a. [1] Substitution

[2] Definition of congruent angles

c. [1] Angle Addition Postulate

[2] Definition of equality

b. [1] Angle Addition Postulate

[2] Definition of congruent angles

d. [1] Substitution

[2] Definition of equality

Name: ________________________ ID: A

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____ 32. Use the given two-column proof to write a paragraph proof.

Given: ∠1 and ∠2 are supplementary. ∠1 ≅ ∠2. ∠2 ≅ ∠3.

Prove: ∠3 is a right angle.

Two-column proof:

Statements Reasons

1. ∠1 and ∠2 are supplementary.

∠1 ≅ ∠2. ∠2 ≅ ∠3.

1. Given

2. ∠1 and ∠2 are right angles. 2. Congruent supplementary angles form

right angles.

3. m∠2 = 90° 3. Definition of a right angle

4. m∠2 = m∠3 4. Definition of congruent angles

5. m∠3 = 90° 5. Substitution

6. ∠3 is a right angle. 6. Definition of a right angle

Complete the proof.

Paragraph proof:

∠2 ≅ ∠3 is a given statement. Since [1], ∠1 and ∠2 are right angles. By the definition of a right angle,

m∠2 = 90°. By the definition of congruent angles, [2]. Then, m∠3 = 90° by substitution. Therefore, ∠3 is a

right angle by the definition of a right angle.

a. [1] ∠1 and ∠2 are congruent angles

[2] m∠1 = m∠3

b. [1] by definition of a right angle

[2] m∠1 = m∠3

c. [1] congruent, supplementary angles form right angles

[2] m∠2 = m∠3

d. [1] m∠1 = 90° and m∠3 = 90°

[2] m∠2 = m∠3

____ 33. 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°

Name: ________________________ ID: A

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____ 34. Use p and q to find the truth value of the compound statement p ∧ q.

p : Blue is a color.

q : The sum of the measures of the angles of a triangle is 160°.

a. Since p is true, the conjunction is true.

b. Since q is true, the conjunction is true.

c. Since p and q are true, the conjunction is true.

d. Since q is false, the conjunction is false.

____ 35. Construct a truth table for the compound statement ∼ a ∧ ∼ b.

a.

a b ∼ a ∼ b ∼ a ∧ ∼ b

T T F F T

T F F T F

F T T F F

F F T T T

c.

a b ∼ a ∼ b ∼ a ∧ ∼ b

T T T T F

T F T F F

F T F T F

F F F F T

b.

a b ∼ a ∼ b ∼ a ∧ ∼ b

T T T T T

T F T F F

F T F T F

F F F F F

d.

a b ∼ a ∼ b ∼ a ∧ ∼ b

T T F F F

T F F T F

F T T F F

F F T T T

ID: A

1

Chapter 2 review test

Answer Section

MULTIPLE CHOICE

1. ANS: A

The prime numbers make up the pattern. The next prime is 13.

Feedback

A Correct!

B The prime numbers make up the pattern. What is the next prime number?

C The prime numbers make up the pattern. What is the next prime number?

D 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 STA: (G.5)(B) TOP: 2-1 Using Inductive Reasoning to Make Conjectures

2. ANS: A

List some examples and look for a pattern.

3 + 5 = 8 3 + 7 = 10 5 + 7 = 12 5 + 9 = 14

Feedback

A Correct!

B List some examples and look for a pattern.

C List some examples and look for a pattern.

D List some examples and look for a pattern.

PTS: 1 DIF: Basic REF: Page 74 OBJ: 2-1.2 Making a Conjecture

NAT: 12.3.5.a STA: (G.3)(D) TOP: 2-1 Using Inductive Reasoning to Make Conjectures

3. 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 Correct!

B Look at the table's data. Are there more men or women 65 years and over?

C Look at the table's data. Is there any information about previous years?

D 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 STA: (G.2)(B) TOP: 2-1 Using Inductive Reasoning to Make Conjectures

ID: A

2

4. 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 > ba

b> 0 Conclusion

Let a = 4 and b = 1. 4 > 14

1> 0 The conjecture holds.

Let a = 11 and b = 3. 11 > 311

3> 0 The conjecture holds.

Let a = 11 and b = −3. 11 > −311

−3< 0 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 Correct!

B In this case, a/b is greater than zero, so it is not a counterexample.

C In this case, a is not greater than b. The counterexample should have a > b and a/b less

than or equal to 0.

D 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 STA: (G.3)(C) TOP: 2-1 Using Inductive Reasoning to Make Conjectures

5. ANS: A

x values 1 2 3 4 5 6

x2

− 16x + 64 49 36 25 16 9 4

The pattern appears to be a decreasing set of perfect squares.

When x = 7, 72

− 16(7) + 64 = 1. This follows the pattern.

When x = 8, 82

− 16(8) + 64 = 0. This follows the pattern.

When x = 9, 92

− 16(9) + 64 = 1. This does not follow the pattern.

Thus, x = 9 generates a counterexample.

Feedback

A Correct!

B 0 is a perfect square. Try higher values of x.

C 1 is a perfect square. Try higher values of x.

D 0 is a perfect square. Try higher values of x.

PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: (G.3)(C)

TOP: 2-1 Using Inductive Reasoning to Make Conjectures

ID: A

3

6. ANS: A

For an if-then conditional statement, the hypothesis is the part following the word if.

Hypothesis: It is raining.

Conclusion: It is cloudy.

Feedback

A Correct!

B The hypothesis is the part of the statement following the word "if."

C This is not part of the statement. The hypothesis is the part of the statement following

the word "if."

D The hypothesis is the part of the statement following the word "if."

PTS: 1 DIF: Basic REF: Page 81

OBJ: 2-2.1 Identifying the Parts of a Conditional Statement NAT: 12.3.5.a

TOP: 2-2 Conditional Statements

7. ANS: C

Identify the hypothesis and conclusion.

Hypothesis Conclusion

A horse has 4 legs.

If it is a horse, then it has 4 legs.

Feedback

A Identify the hypothesis and conclusion.

B A conditional statement should have a hypothesis and a conclusion.

C Correct!

D A conditional statement should have a hypothesis and a conclusion.

PTS: 1 DIF: Average REF: Page 82

OBJ: 2-2.2 Writing a Conditional Statement NAT: 12.3.5.a

STA: (G.3)(B) TOP: 2-2 Conditional Statements

8. ANS: B

There are several figures with four sides that are not squares.

So, the conditional statement is false.

Counterexample: A rectangle has four sides, and it is not a square.

Feedback

A There is a counterexample.

B Correct!


OBJ: 2-2.3 Analyzing the Truth Value of a Conditional Statement

NAT: 12.3.5.a STA: (G.3)(A) TOP: 2-2 Conditional Statements

ID: A

4

9. ANS: B

Conditional: p → q If an animal is a bird, then it has two eyes.

Converse: q → p If an animal has two eyes, then it is a bird.

Inverse: ∼ p → ∼ q If an animal is not a bird, then it does not have two eyes.

Contrapositive: ∼ q → ∼ p If an animal does not have two eyes, then it is not a bird.

Feedback

A To find the contrapositive, exchange and negate the hypothesis and the conclusion.

B Correct!

C To find the inverse, negate the hypothesis and the conclusion.

D To find the converse, exchange the hypothesis and the conclusion.

PTS: 1 DIF: Average REF: Page 83 OBJ: 2-2.4 Application

NAT: 12.3.5.a STA: (G.3)(C) TOP: 2-2 Conditional Statements

ID: A

5

10. ANS: B

Create a Venn diagram that represents the set of rational numbers, integers, and whole numbers.

A conditional statement will be true when the set referred to in the hypothesis is a subset of the set referred to

in the conclusion.

If n is a whole number, then n is an integer.

If n is a whole number, then n is a rational number.

If n is an integer, then n is a rational number.

You can write three true conditional statements using the statements given.

Feedback

A Create a Venn diagram relating the set of rational numbers, integers, and whole

numbers.

B Correct!

C Create a Venn diagram relating the set of rational numbers, integers, and whole

numbers.

D Create a Venn diagram relating the set of rational numbers, integers, and whole

numbers.

PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: (G.3)(A)

TOP: 2-2 Conditional Statements

11. 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 Correct!

B Deductive reasoning is based on logical reasoning.

C Inductive reasoning is based on observation.

D 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 STA: (G.3)(C) TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

ID: A

6

12. ANS: D

Identify the hypothesis and conclusion in the given conditional.

Hypothesis: Tommy makes cookies tonight.

Conclusion: Tommy must have an oven.

The given statement, “Tommy has an oven,” matches the conclusion, but that does not mean that the

hypothesis is true. Tommy could have an oven but he could use it for something besides cookies.

The conjecture is not valid.

Feedback

A By the Law of Detachment, if the conditional and hypothesis are valid, then the

conclusion is valid.

B This statement does not show that the original conjecture is not valid.

C By the Law of Detachment, if the conditional and hypothesis are valid, then the

conclusion is valid.

D Correct!


OBJ: 2-3.2 Verifying Conjectures by Using the Law of Detachment

NAT: 12.3.5.a STA: (G.3)(E) TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

13. 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 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.

B Correct!


OBJ: 2-3.3 Verifying Conjectures by Using the Law of Syllogism

NAT: 12.3.5.a STA: (G.3)(E) TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

ID: A

7

14. ANS: D

Two lines meet at a 90° angle.

It given that if two lines meet at a 90° angle, then they are perpendicular.

It is also given that if two lines are perpendicular, then they form right angles.

So, the conclusion is: The lines form a right angle.

Feedback

A Find the end conclusion.

B Start with the given information that two lines meet at a 90-degree angle.

C Find the end conclusion.

D Correct!


OBJ: 2-3.4 Applying the Laws of Deductive Reasoning NAT: 12.3.5.a

STA: (G.3)(E) TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

15. ANS: C

The conclusion from the given conditional statements is, if you are eating a banana, then you are eating food.

The contrapositive of the first statement is, if you are not eating fruit, then you are not eating a banana.

The contrapositive of the second statement is, if you are not eating food, then you are not eating fruit.

The conclusion from the contrapositives is, if you are not eating food, then you are not eating a banana.


Feedback

A If "p implies q" is a conditional, then its contrapositive is "not q implies not p."

B If "p implies q" is a conditional, then its contrapositive is "not q implies not p."

C Correct!

D If p implies q and q implies r then p implies r.

PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: (G.3)(A)

TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

ID: A

8

16. ANS: A

Let p and q represent the following.

p: A rectangle is a square.

q: All four sides of the rectangle are equal lengths.

The two parts of the biconditional p ↔ q are p → q and q → p.

Conditional: If all four sides of the rectangle are equal length, then it is a square.

Converse: If a rectangle is a square, then its four sides are equal length.

Feedback

A Correct!

B You can break apart the biconditional statement into a conditional statement and its

converse.

C You can break apart the biconditional statement into a conditional statement and its

converse.

D You can break apart the biconditional statement into a conditional statement and its

converse.


OBJ: 2-4.1 Identifying the Conditionals within a Biconditional Statement

NAT: 12.3.5.a STA: (G.3)(B) TOP: 2-4 Biconditional Statements and Definitions

17. ANS: B

Let p and q represent the following.

p: It is a right triangle.

q: a2

+ b2

= c2.

The given conditional is p → q.

The converse is q → p. If a2

+ b2

= c2, then the figure is a right triangle with sides a, b, and c.

The biconditional is p ↔ q. A figure is a right triangle with sides a, b, and c if and only if a2

+ b2

= c2.

Feedback

A Find the converse, not inverse.

B Correct!

C Find the converse, not the contrapositive.

D Find the converse, not the contrapositive.


OBJ: 2-4.2 Writing a Biconditional Statement NAT: 12.3.5.a

STA: (G.3)(B) TOP: 2-4 Biconditional Statements and Definitions

ID: A

9

18. ANS: B

Conditional: If a figure is a square, then it is a rectangle.

True.

Converse: If a figure is a rectangle, then it is a square.

False. A rectangle does not necessarily have four congruent sides.

Because the converse is false, the biconditional is false.

Feedback

A For a biconditional statement to be true, both the conditional statement and its converse

must be true.

B Correct!

C All rectangles have four 90-degree angles as well.

D A rectangle does have four 90-degree angles, but does it have four congruent sides?


OBJ: 2-4.3 Analyzing the Truth Value of a Biconditional Statement

NAT: 12.3.5.a STA: (G.3)(A) TOP: 2-4 Biconditional Statements and Definitions

19. ANS: B

Think of the definition as being reversible.

Let p be ‘an angle is acute.’

Let q be ‘its measure is less than 90°.’

Conditional: If an angle is acute, then its measure is less than 90°.

Converse: If an angle’s measure is less than 90°, then it is acute.

Biconditional: An angle is acute if and only if its measure is less than 90°.

Feedback

A A biconditional statement uses "if and only if."

B Correct!

C A biconditional statement uses "if and only if."

D Use information from the definition and "if and only if" to form the biconditional.


OBJ: 2-4.4 Writing Definitions as Biconditional Statements NAT: 12.3.5.a

STA: (G.3)(B) TOP: 2-4 Biconditional Statements and Definitions

ID: A

10

20. ANS: A

The conditional statement is the definition of a midpoint, and is a true statement.

The converse is false. The picture displays a counterexample.

AB = BC, but B is not on AC→←

. Therefore, B is not the midpoint of AB. If either the conditional or the

converse is false, the biconditional is false.

Feedback

A Correct!

B Think of possible counterexamples.

C One of the statements from the biconditional is an actual definition.

D One of the statements from the biconditional is an actual definition.

PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: (G.3)(B)

21. ANS: B

4x − 6 = 34 Given equation

+6 +6 [1] Addition Property of Equality

4x = 40 Simplify.

4x

4=

40

4[2] Division Property of Equality

x = 10 Simplify.

Feedback

A Check the properties.

B Correct!

C Check the properties.

D Check the properties.


OBJ: 2-5.1 Solving an Equation in Algebra NAT: 12.5.2.e

STA: (G.3)(E) TOP: 2-5 Algebraic Proof

ID: A

11

22. ANS: A

P = 2l + 2w Given equation26 = 2(8) + 2w [1] Substitution Property of Equality

26 = 16 + 2w

−16 = −16

10 = 2w

Simplify.

Subtraction Property of Equality

Simplify.

10

2=

2w

2[2] Division Property of Equality

5 = w Simplify.

w = 5 Symmetric Property of Equality

Feedback

A Correct!

B The variables P and l are substituted, not simplified. Use the Substitution Property.

C Check the properties.

D Check the justifications.

PTS: 1 DIF: Average REF: Page 105 OBJ: 2-5.2 Problem-Solving Application

NAT: 12.5.2.e STA: (G.3)(E) TOP: 2-5 Algebraic Proof

23. ANS: B

m∠JKL = m∠JKM + m∠MKL [1] Angle Addition Postulate

100° = (6x + 8)° + (2x − 4)° Substitution Property of Equality

100 = 8x + 4 Simplify.

96 = 8x Subtraction Property of Equality

12 = x [2] Division Property of Equality

x = 12 Symmetric Property of Equality

Feedback

A Check the properties.

B Correct!

C Check the justifications.

D The Segment Addition Postulate refers to segments, not angles.


OBJ: 2-5.3 Solving an Equation in Geometry NAT: 12.5.2.e

STA: (G.3)(B) TOP: 2-5 Algebraic Proof

ID: A

12

24. ANS: D

The Transitive Property of Congruence states that if figure A ≅ figure B and figure B ≅ figure C, then figure

A ≅ figure C.

Feedback

A The Reflexive Property of Congruence states that figure A is congruent to figure A.

B The Substitution Property of Equality states that if a = b, then b can be substituted for a

in any expression.

C The Symmetric Property of Congruence states that if figure A is congruent to figure B,

then figure B is congruent to figure A.

D Correct!


OBJ: 2-5.4 Identifying Properties of Equality and Congruence NAT: 12.5.2.e

STA: (G.3)(B) TOP: 2-5 Algebraic Proof

25. ANS: D

EG = FH Given information

EG = EF + FG Segment Addition Postulate

FH = FG + GH Segment Addition Postulate

EF + FG = FG + GH Substitution Property of Equality

EF = GH Subtraction Property of Equality

Feedback

A The Angle Addition Postulate refers to angles, not segments.

B Check the properties.

C Check the steps.

D Correct!

PTS: 1 DIF: Average REF: Page 110 OBJ: 2-6.1 Writing Justifications

NAT: 12.3.5.a STA: (G.3)(B) TOP: 2-6 Geometric Proof

ID: A

13

26. ANS: C

Proof:

Statements Reasons

1. ∠1 and ∠2 are supplementary. 1. Given

2. m∠1 = 135° 2. Given

3. m∠1 + m∠2 = 180° 3. Definition of supplementary angles

4. 135° + m∠2 = 180° 4. Substitution Property

5. m∠2 = 45° 5. Subtraction Property of Equality

Feedback

A Check to the given information.

B To get from Step 4 to Step 5, use subtraction, not substitution.

C Correct!

D The angles are supplementary, not complementary.


OBJ: 2-6.2 Completing a Two-Column Proof NAT: 12.3.5.a

STA: (G.1)(A) TOP: 2-6 Geometric Proof

27. ANS: A

Proof:

Statements Reasons

1. m∠1 + m∠2 = 90° 1. Given

2. m∠3 + m∠4 = 90° 2. Given


4. m∠2 = m∠3 4. Given


6. m∠1 = m∠4 6. Subtraction Property of Equality

Feedback

A Correct!

B Check the given information.

C To get from Step 4 to Step 5, use substitution, not subtraction.

D To get from Step 4 to Step 5, use substitution, not addition.


OBJ: 2-6.3 Writing a Two-Column Proof from a Plan NAT: 12.3.5.a

STA: (G.3)(E) TOP: 2-6 Geometric Proof

ID: A

14

28. ANS: B

Step 1 Create an equation

The angles are supplements and their sum equals 180°.

(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 = −32

3or 6.

When x = −32

3, 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

The measurement of the first angle is 2x2

+ 3x − 5 = 2(6)2

+ 3(6) − 5 = 85°.

The measurement of the second angle is x2

+ 11x − 7 = (6)2

+ 11(6) − 7= 95°.

Feedback

A The angles are supplements. Use the definition of supplements to solve for x.

B Correct!

C Check for algebra mistakes. When x equals 5, the second angle is not 120 degrees.

D The angles are supplements. Use the definition of supplements to solve for x.

PTS: 1 DIF: Advanced NAT: 12.2.1.f TOP: 2-6 Geometric Proof

29. ANS: D

In a flowchart, reasons flow from the statement above. The statement above Reason 2 is AB + BF = FC + CD.

The statement above Reason 3 is AB + BF = AF ; FC + CD = FD.

Feedback

A Reasons flow from the statement above.

B Reasons flow from the statement above.

C Reasons flow from the statement above.

D Correct!

PTS: 1 DIF: Average REF: Page 118 OBJ: 2-7.1 Reading a Flowchart Proof

NAT: 12.3.5.a STA: (G.1)(A) TOP: 2-7 Flowchart and Paragraph Proofs

ID: A

15

30. ANS: B

In a flowchart, reasons follow statements. Using the two-column proof, the statement that leads to Reason 2

is ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary. The reason that follows Statement 3 is

Congruent Supplements Theorem.

Feedback

A Angles 1 and 2 are supplements, not complements.

B Correct!

C In a flowchart, reasons follow statements.

D In a flowchart, reasons follow statements.

PTS: 1 DIF: Average REF: Page 119 OBJ: 2-7.2 Writing a Flowchart Proof

NAT: 12.3.5.a STA: (G.3)(E) TOP: 2-7 Flowchart and Paragraph Proofs

31. ANS: B

Two-column proof:

Statements Reasons

1. ∠BAC is a right angle. ∠1 ≅ ∠3 1. Given

2. m∠BAC = 90° 2. Definition of a right angle

3. m∠BAC = m∠1 + m∠2 3. Angle Addition Postulate


5. m∠1 = m∠3 5. Definition of congruent angles


7. ∠2 and ∠3 are complementary. 7. Definition of complementary angles

Feedback

A In a paragraph proof, statements and reasons appear together.

B Correct!

C In a paragraph proof, statements and reasons appear together.

D In a paragraph proof, statements and reasons appear together.

PTS: 1 DIF: Average REF: Page 120 OBJ: 2-7.3 Reading a Paragraph Proof

NAT: 12.3.5.a STA: (G.1)(A) TOP: 2-7 Flowchart and Paragraph Proofs

32. ANS: C

In a paragraph proof, statements and reasons appear together. The reason following the statement, “∠1 and

∠2 are right angles,” is “congruent, supplementary angles form right angles.” The statement preceding the

reason, “Definition of congruent angles,” is “m∠2 = m∠3.”

Feedback

A In a paragraph proof, statements and reasons appear together.

B In a paragraph proof, statements and reasons appear together.

C Correct!

D In a paragraph proof, statements and reasons appear together.

PTS: 1 DIF: Average REF: Page 121 OBJ: 2-7.4 Writing a Paragraph Proof

NAT: 12.3.5.a STA: (G.3)(E) TOP: 2-7 Flowchart and Paragraph Proofs

ID: A

16

33. ANS: D

Step 1 Create a system of equations.

m∠1 = m∠320x + 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

Multiply the first equation by −2.

Multiply the second equation by 7.

131x = 917 Add the two equations together.

x = 7 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 Use the definitions of supplementary and vertical angles to create a solvable system of

equations.

B Use the definitions of supplementary and vertical angles to create a solvable system of

equations.

C Use the definitions of supplementary and vertical angles to create a solvable system of

equations.

D Correct!

PTS: 1 DIF: Advanced NAT: 12.2.1.f TOP: 2-7 Flowchart and Paragraph Proofs

ID: A

17

34. ANS: D

A conjunction is true only when all of its parts are true.

p is true. q is false. So p ∧ q is false.

Feedback

A A conjunction is true only when all of its parts are true.

B A conjunction is true only when all of its parts are true.

C Check that both p and q are true.

D Correct!


OBJ: 2-Ext.1 Analyzing Truth Values of Conjunctions and Disjunctions

TOP: 2-Ext Introduction to Symbolic Logic

35. ANS: D

Since a and b can each be either true or false, the truth table will have (2)(2) = 4 rows. The negation (∼) of a

statement has the opposite truth value.

a b ∼ a ∼ b ∼ a ∧ ∼ b

T T F F F

T F F T F

F T T F F

F F T T T

Feedback

A Check the values for "~a AND ~b".

B The negation of a statement has the opposite truth value, not the same truth value.

C Check the values for "~a" and "~b".

D Correct!


OBJ: 2-Ext.2 Constructing Truth Tables for Compound Statements

TOP: 2-Ext Introduction to Symbolic Logic

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