Geometry Study Guide - Weeblywhamrsm.weebly.com/uploads/8/2/6/6/8266221/chapter_2_study_guide.pdf37. Use the Law of Syllogism to draw a conclusion from the given information. Given:

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

1

Geometry Study Guide

Matching

Match each vocabulary term with its definition.

a. conjecture e. biconditional statement

b. inductive reasoning f. hypothesis

c. deductive reasoning g. counterexample

d. conclusion h. conditional statement

____ 1. an example that proves that a conjecture or statement is false

____ 2. a statement that is believed to be true

____ 3. the part of a conditional statement following the word then

____ 4. the part of a conditional statement following the word if

____ 5. the process of reasoning that a rule or statement is true because specific cases are true

____ 6. a statement that can be written in the form “if p, then q,” where p is the hypothesis and q is the conclusion


a. conclusion e. hypothesis

b. converse f. truth value

c. inverse g. contrapositive

d. negation

____ 7. for a statement, either true (T) or false (F)

____ 8. operations that undo each other

____ 9. the contradiction of a statement by using “not,” written as ∼

____ 10. the statement formed by exchanging the hypothesis and conclusion of a conditional statement

____ 11. the statement formed by both exchanging and negating the hypothesis and conclusion


a. logically equivalent statements f. quadrilateral

b. deductive reasoning g. pentagon

c. biconditional statement h. definition

d. inductive reasoning i. triangle

e. polygon

____ 12. a statement that describes a mathematical object and can be written as a true biconditional statement

____ 13. statements that have the same truth value

____ 14. a four-sided polygon

____ 15. a closed plane figure formed by three or more segments such that each segment intersects exactly two other

segments only at their endpoints and no two segments with a common endpoint are collinear

____ 16. the process of using logic to draw conclusions

____ 17. a statement that can be written in the form “p if and only if q”

____ 18. a three-sided polygon

Name: ________________________ ID: A

2


a. deductive reasoning e. inductive reasoning

b. paragraph proof f. two-column proof

c. proof g. flowchart proof

d. theorem

____ 19. a style of proof in which the statements are written in the left-hand column and the reasons are written in the

right-hand column

____ 20. a statement that has been proven

____ 21. a style of proof in which the statements and reasons are presented in paragraph form

____ 22. an argument that uses logic to show that a conclusion is true

____ 23. a style of proof that uses boxes and arrows to show the structure of the proof

Short Answer

24. Find the next item in the pattern 2, 3, 5, 7, 11, ...

25. Complete the conjecture.

The sum of two odd numbers is ____.

26. 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

27. Show that the conjecture is false by finding a counterexample.

If a > b, then a

b> 0.

28. 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?

29. Identify the hypothesis and conclusion of the conditional statement.

If it is raining then it is cloudy.

30. Write a conditional statement from the statement.

A horse has 4 legs.

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

it is a square.

32. Write the converse, inverse, and contrapositive of the conditional statement.

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

Name: ________________________ ID: A

3

33. 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.

34. 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?

35. 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.

36. 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.

37. Use the Law of Syllogism to 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.

38. 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.

39. Write the conditional statement and converse within the biconditional.

A rectangle is a square if and only if all four sides of the rectangle have equal lengths.

40. 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.

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

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

42. Write the definition as a biconditional.

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

43. 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.

44. 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.

Name: ________________________ ID: A

4

45. 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

46. 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

47. Identify the property that justifies the statement.

AB ≅ CD and CD ≅ EF . So AB ≅ EF .

48. 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

Name: ________________________ ID: A

5

49. 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]

Name: ________________________ ID: A

6

50. 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]

51. Two angles with measures (2x2

+ 3x − 5)° and (x2

+ 11x − 7)° are supplementary. Find the value of x and the

measure of each angle.

Name: ________________________ ID: A

7

52. 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

Name: ________________________ ID: A

8

53. 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

Name: ________________________ ID: A

9

54. 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

Name: ________________________ ID: A

10

55. 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.

56. 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.

ID: A

1

Geometry Study Guide

Answer Section

MATCHING

1. ANS: G PTS: 1 DIF: Basic REF: Page 75

TOP: 2-1 Using Inductive Reasoning to Make Conjectures

2. ANS: A PTS: 1 DIF: Basic REF: Page 74


3. ANS: D PTS: 1 DIF: Basic REF: Page 81

TOP: 2-2 Conditional Statements

4. ANS: F PTS: 1 DIF: Basic REF: Page 81


5. ANS: B PTS: 1 DIF: Basic REF: Page 74


6. ANS: H PTS: 1 DIF: Basic REF: Page 81




8. ANS: C PTS: 1 DIF: Basic REF: Page 83








12. ANS: H PTS: 1 DIF: Basic REF: Page 97

TOP: 2-4 Biconditional Statements and Definitions

13. ANS: A PTS: 1 DIF: Basic REF: Page 83




15. ANS: E PTS: 1 DIF: Basic REF: Page 98



TOP: 2-3 Using Deductive Reasoning to Verify Conjectures



18. ANS: I PTS: 1 DIF: Basic REF: Page 98



TOP: 2-6 Geometric Proof

ID: A

2




TOP: 2-7 Flowchart and Paragraph Proofs


TOP: 2-5 Algebraic Proof


TOP: 2-7 Flowchart and Paragraph Proofs

SHORT ANSWER

24. ANS:

13

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

PTS: 1 DIF: Basic REF: Page 74 OBJ: 2-1.1 Identifying a Pattern

NAT: 12.5.1.a STA: GE1.0 TOP: 2-1 Using Inductive Reasoning to Make Conjectures

25. ANS:

even

List some examples and look for a pattern.

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

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


26. ANS:

Women outnumbered men in the 65 years and over population.

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.

PTS: 1 DIF: Average REF: Page 75 OBJ: 2-1.3 Application


ID: A

3

27. ANS:

a = 11, b = −3

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.

PTS: 1 DIF: Average REF: Page 76 OBJ: 2-1.4 Finding a Counterexample


28. ANS:

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

x = 9 generates a counterexample.

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.

PTS: 1 DIF: Advanced NAT: 12.3.5.a STA: GE3.0


29. ANS:

Hypothesis: It is raining.

Conclusion: It is cloudy.

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

Hypothesis: It is raining.

Conclusion: It is cloudy.

PTS: 1 DIF: Basic REF: Page 81

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

STA: GE3.0 TOP: 2-2 Conditional Statements

ID: A

4

30. ANS:

If it is a horse then it has 4 legs.

Identify the hypothesis and conclusion.

Hypothesis Conclusion

A horse has 4 legs.

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

PTS: 1 DIF: Average REF: Page 82

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

STA: GE3.0 TOP: 2-2 Conditional Statements

31. ANS:

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

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.


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

NAT: 12.3.5.a STA: GE3.0 TOP: 2-2 Conditional Statements

32. ANS:

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.

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.

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

NAT: 12.3.5.a STA: GE3.0 TOP: 2-2 Conditional Statements

ID: A

5

33. ANS:

3 conditional statements

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.



34. ANS:

Since the conclusion is based on a pattern of observation, it is a result of inductive reasoning.

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.

PTS: 1 DIF: Basic REF: Page 88 OBJ: 2-3.1 Application

NAT: 12.3.5.a STA: GE3.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

35. ANS:

The conjecture is not valid, because Tommy could have an oven but he could make something besides

cookies tonight.

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.


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


ID: A

6

36. ANS:

Yes, the conjecture is valid.

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.


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


37. ANS:

Conclusion: The lines form a right angle.

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.


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

STA: GE1.0 TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

38. ANS:

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.

The second conclusion is the contrapositive of the first conclusion.

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.

The second conclusion is the contrapositive of the first conclusion.


TOP: 2-3 Using Deductive Reasoning to Verify Conjectures

ID: A

7

39. ANS:

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

Converse: If a rectangle is a square, then its four sides have equal lengths.

Let p and q represent the following.

p: A rectangle is a square.

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

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

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

Converse: If a rectangle is a square, then its four sides have equal lengths.


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

NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions

40. ANS:

Converse: If a2

+ b2

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

Biconditional: A figure is a right triangle with sides a, b, and c if and only if a2

+ b2

= c2.

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.


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

STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions

41. ANS:

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

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.


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

NAT: 12.3.5.a STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions

ID: A

8

42. ANS:

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

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°.


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

STA: GE3.0 TOP: 2-4 Biconditional Statements and Definitions

43. ANS:

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.

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.



ID: A

9

44. ANS:

[1] Addition Property of Equality;

[2] Division Property of Equality4x − 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.


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

STA: 1A5.0 TOP: 2-5 Algebraic Proof

45. ANS:

[1] Substitution Property of Equality

[2] Division Property of Equality

The garden is 5 ft wide.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

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

NAT: 12.5.2.e STA: 1A5.0 TOP: 2-5 Algebraic Proof

46. ANS:

[1] Angle Addition Postulate

[2] Division Property of Equality

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


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

STA: GE1.0 TOP: 2-5 Algebraic Proof

ID: A

10

47. ANS:

Transitive Property of Congruence

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

A ≅ figure C.


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

STA: GE4.0 TOP: 2-5 Algebraic Proof

48. ANS:

[1] Segment Addition Postulate

[2] Substitution Property of Equality

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

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

NAT: 12.3.5.a STA: GE2.0 TOP: 2-6 Geometric Proof

49. ANS:

[1] m∠1 = 135°

[2] Definition of supplementary angles

[3] Subtraction Property of Equality

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


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

STA: GE2.0 TOP: 2-6 Geometric Proof

ID: A

11

50. ANS:

[1] m∠3 + m∠4 = 90°

[2] Substitution Property

[3] Subtraction Property of Equality

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


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

STA: GE2.0 TOP: 2-6 Geometric Proof

51. ANS:

x = 6; 85°; 95°

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°.

PTS: 1 DIF: Advanced NAT: 12.2.1.f STA: 6MG2.2


52. ANS:

[1] AB + BF = FC + CD

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

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.

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

NAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs

ID: A

12

53. ANS:

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

[2] Congruent Supplements Theorem

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.

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


54. ANS:

[1] Angle Addition Postulate

[2] Definition of congruent angles

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

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


55. ANS:

[1] congruent, supplementary angles form right angles

[2] m∠2 = m∠3

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.”

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


ID: A

13

56. ANS:

x = 7; y = 9; 147°; 147°; 33°; 33°

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 = 18020x + 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°.

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

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