Geometry: A Complete Course · are opposite rays, then the angles are supplementary.” For each of the following statements 1 through 10, write either true or false. 1. Two angles

Written by: Larry E. Collins

Geometry:A Complete Course

(with Trigonometry)

Module C – Instructor's Guidewith Detailed Solutions for

Progress Tests

Videotext3

Errata 10/2008

Name

Class Date Score

Quiz Form A

Unit III - Fundamental TheoremsPart A - Deduction ProofLesson 2 - InDirect Proof

1. State the negation of each statement

a) It will not rain. ___________________________________________________________________

b) nABC is an isosceles triangle. ______________________________________________________

c) “x + 5” is not an open phrase. _______________________________________________________

2. Indicate whether each pair of statements would enable you to arrive at a contradiction in an indirect proof, and give some justification for your answer.

a) AB < 15; AB > 20 ________________________________________________________________

________________________________________________________________________________

b) /X and /Y are obtuse angles; /X and /Y are supplementary. ___________________________

________________________________________________________________________________

c) Point B is between points A and C; Points A, B, and C are not collinear. _____________________

________________________________________________________________________________

d) /P and /Q are congruent; /P and /Q are complementary. ______________________________

________________________________________________________________________________

3. For each of the following conditionals, state the assumption you would use to start an indirect proof.

a) If a triangle is equilateral, then the triangle is isosceles. __________________________________

_______________________________________________________________________________

b) If an angle is a right angle, then the angle is equal to its supplement. _______________________

_______________________________________________________________________________

© 2006 VideoTextInteractive Geometry: A Complete Course 9

It is not true that it will not rain. Or, it will rain.

It is not true that nABC is an isoceles triangle. Or, nABC is not an isoceles triangle.

It is not true that “x + 5” is not an open phrase. Or “x + 5” is an open phrase.

Yes. These two statements are already contradictory

Yes. /X and /Y are greater than 90.

Yes, by the definition

No. Both can be true without contradiction

Suppose (assume) the triangle is not isoceles.

Suppose (assume) the angle is

not equal to its supplement.

of Betweenness


Name

Class Date Score

Quiz Form A

Unit III - Fundamental TermsPart D - Theorems About Segments and RaysLesson 1 - Theorem 5: “If two different lines intersect, then exactlyone plane contains both lines.”Lesson 2 - Theorem 6: “If in a plane, there is a point on a line, thenthere is exactly one point perpendicular to the line, through that point.”

1. In the diagram to the right, BE AC and BD BF. Find the measure of each of the following angles.

a) m/EBF ________ b) m/DBE ________

c) m/DBA ________ d) m/DBC ________

2. In the diagram to the right, BE AC and BD BF. Also assume m/CBF = x. Express the measure of each of the following angles:

a) m/EBF ________ b) m/DBE ________

c) m/DBA ________ d) m/DBC ________

3. In the diagram to the right, BE AC and BD BF. Find the value of x in each of the following problems.a) m/DBE = 3x, m/EBF = 4x–1 x = ____________

b) m/ABD = 6x, m/DBE = 3x + 9, x = ____________

m/EBF = 4x + 18, m/FBC = 4x

> >

A B C

,

DF

E

35O

A B C

DF

E

X Y Z

S TR

3 4

1 2

FE

M

RS

A O E

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B

F

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C

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x

y A B C

DF

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DB

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CA

12 4

3

z

A B C

,

DF

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35O

A B C

DF

E

X Y Z

S TR

3 4

1 2

FE

M

RS

A O E

D

B

F

H

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x

y A B C

DF

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12 4

3

z

90 – 35 = 55 90 – 55 = 35

90 – 35 = 55 90 + 35 = 125

90 – (90 – x) or x

3x + (4x – 1) = 907x – 1 = 90

7x = 91x = 13

6x + 3x + 9 + 4x + 18 + 4x = 18017x + 27 = 180

17x = 153x = 9

55O 35O

55O 125O

90 – x x

90 + x90 – x

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© 2006 VideoTextInteractive Geometry: A Complete Course26

Unit III, Part D, Lessons 1&2, Quiz Form A—Continued—

Name

4. Using the format given below, write a complete proof with the

“Given” and “Prove” information, and the “Diagram”.

Given: m/1 = m/2 and Diagram:

m/3 = m/4 as shown

Prove: YS XZ

1. m/1 = m/2 1. Given

2. m/3 = m/4 2. Given

3. m/1 + m/3 = m/2 + m/4 3. Addition for Equality

4. m/1 + m/3 = m/XYS 4. Postulate 7 (Protractor) - Angle-Addition Assumption

5. m/2 + m/4 = m/SYZ 5. Postulate 7 (Protractor) - Angle-Addition Assumption

6. m/XYS = m/SYZ 6. Substitution (from statements 3, 4, and 5)

7. /XYZ is a straight angle whose 7. Definition of Straight Anglemeasure of 180O

8. m/XYS + m/SYZ = m/XYZ 8. Postulate 6 (Ruler) - Angle-Addition Assumption

9. m/XYS + m/SYZ = 180 9. Substitution (from statements 7 and 8)

10. m/XYS + m/XYS = 180 10. Substitution (from statements 6 and 9)

11. 2m/XYS = 180 11. Collect like terms (Distributivity)

12. /XYS = 90 12. Multiplication for Equality

13. /XYS is a right / 13. Definition of Right Angle

14. YS XZ 14. Definition of Perpendicular Lines

A B C

,

DF

E

35O

A B C

DF

E

X Y Z

S TR

3 4

1 2

FE

M

RS

A O E

D

B

F

H

C

G

x

y A B C

DF

E

P

C

D

BA

DB

E

CA

12 4

3

z

>

>

STATEMENT REASON


Name

Class Date Score

Quiz Form A

Unit III - Fundamental TermsPart F - Theorems About Segments and RaysLesson 1 - Theorem 9: “If two adjacent acute angles have their exteriorsides in perpendicular lines, then the two angles are complementary.”Lesson 2 - Theorem 10: “If the exterior sides of two adjacent anglesare opposite rays, then the angles are supplementary.”

For each of the following statements 1 through 10, write either true or false.

1. Two angles may be both adjacent and congruent. _______________

2. Two angles may be both complementary and supplementary. _______________

3. If two angles are acute, they cannot be supplementary. _______________

4. If two lines intersect, then four pairs of supplementary and adjacent angles are formed. _______________

5. If /AOB and /BOC are supplementary and adjacent, then OAand OC cannot be a pair of opposite rays. _______________

6. If /AOB and /BOC are adjacent, then B lies inside /AOC. _______________

7. If two angles formed by two lines are adjacent, then they are supplementary. _______________

8. If B lies inside /AOC, then /AOB and /BOC are adjacent. _______________

9. If the measure of an angle is 120, the measure of the complement is 60. _______________

10. If /AOB and /BOC are adjacent, then m/AOB + m/BOC = m/AOC. _______________

True

False

True

True

False

True

True

True

False

True

NameUnit III, Part F, Lessons 1&2, Quiz Form A—Continued—


12. Given: /ADC > /BEC as shown

Prove: /BDC > /AEC

1. /ADC and /CDB are adjacent angles 1. Definition of Adjacent Angle/BEC and /CEA are adjacent angles

2. DA and DB are opposite rays 2. Definition of Opposite RaysEB and EA are opposite rays

3. /ADC and /CDB are supplementary 3. Theorem 10 - If the exterior sides of two adjacent angles are/BEC and /CEA are supplementary opposite rays, then the angles are supplementary.

4. m/ADC + m/CDB = 180 4. Definition of Supplementary Anglesm/BEC + m/CEA = 180

5. m/ADC + m/CDB = m/BEC + m/CEA 5. Substitution of Equals (4 into 4)

6. /ADC > /BEC 6. Given

7. m/ADC = m/BEC 7. Definition of Congruent Angles

8. m/ADC + m/CDB – m/ADC = m/BEC + 8. Addition Property of Equalitym/CEA – m/BEC

9. m/ADC + m/CDB + —m/ADC = m/BEC + 9. Definition of Subtraction (a – b means a + b)m/CEA + —m/BEC

10. m/ADC + —m/ADC + m/CDB = m/BEC + 10. Commutative Property of Addition—m/BEC + m/CEA

11. 0 + m/CDB = 0 + m/CEA 11. Additive Inverse Property

12. m/CDB = m/CEA 12. Identity Property of Addition

13. /BDC > /AEC 13. Definition of Congruent Angles

A O

BC D

D E

BA

C

U

V

T

W

X

Y

R

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X

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A

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A

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A

B

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1 2

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DA

E

C

B

DA

CB

E

CA– 4– 5 – 3 – 2 – 1 0 1

– 3– 4 – 2 – 1 0 1 2

STATEMENT REASON


Name

Class Date Score

Quiz Form A

Unit III - Fundamental TermsPart F - Theorems About Segments and RaysLesson 3 - Theorem 11: “If you have right angles, then those right anglesare congruent.”Lesson 4 - Theorem 12: “If you have straight angles, then thosestraight angles are congruent.”

1. Name each of the following using the figure at the right.

a) Two pairs of opposite rays. _______________

b) Two right angles. _______________

c) Two straight angles. _______________

d) Three acute angles. _______________

e) Three obtuse angles. _______________

f) Two points in the exterior of /VYT _______________

g) The sides of /XYV _______________

h) The vertex of all angles. _______________

i) A point in the interior of /TYV _______________

j) An angle which is congruent to /WYV _______________

k) An angle which is congruent to /TYW _______________

YU and YX; YT and YW

/VYW; /VYT

/TYW; /XYU

/TYU; /UYV; /XYW

/XYT; /UYW; /XYV

point X; point W

YX; YV

point Y

point U

/VYT

/XYU

A O

BC D

D E

BA

C

U

V

T

W

X

Y

R

T

NM 31 2

4Q

X

Z

Y

A

C

B DW

21

,

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2

B

A

F

E

DC

OX BD

A

B

CA D

Q

A B

1 2

X Y

DA

E

C

B

DA

CB

E

CA– 4– 5 – 3 – 2 – 1 0 1

– 3– 4 – 2 – 1 0 1 2

NameUnit III, Part F, Lessons 3&4, Quiz Form A—Continued—


5. Given: /XYD is a right angle, /ABW is a right angle,/XYZ is a straight angle, and /ABC is a straight angle as shown

Prove: /DYZ and /WBC are supplementary

1. /XYD is a right angle 1. Given2. /ABW is a right angle 2. Given

3. m/XYD = 90 3. Definition of a Right Angle4. m/ABW = 90 4. Definition of a Right Angle

5. /XYZ is a straight angle 5. Given6. /ABC is a straight angle 6. Given

7. m/XYZ = 180 7. Definition of a Straight Angle8. m/ABC = 180 8. Definition of a Straight Angle

9. m/XYZ = m/XYD + m/DYZ 9. Postulate 7 (Protractor) – Angle-Addition Assumption10. m/ABC = m/ABW + m/WBC 10. Postulate 7 (Protractor) – Angle-Addition Assumption

11. 180 = 90 + m/DYZ 11. Substitution of Equals12. 180 = 90 + m/WBC 12. Substitution of Equals

13. 180 – 90 = 90 + m/DYZ – 90 13. Subtraction Property for Equality14. 180 – 90 = 90 + m/WBC – 90 14. Subtraction Property for Equality

15. 90 = 90 + m/DYZ – 90 15. Substitution of Equals16. 90 = 90 + m/WBC – 90 16. Substitution of Equals

17. 90 = 90 + m/DYZ + –90 17. Definition of Subtraction18. 90 = 90 + m/WBC + –90 18. Definition of Subtraction

19. 90 = 90 + –90 + m/DYZ 19. Commutative Property of Addition20. 90 = 90 + –90 + m/WBC 20. Commutative Property of Additio

21. 90 = 0 + m/DYZ 21. Additive Inverse Property22. 90 = 0 + m/WBC 22. Additive Inverse Property

23. 90 = m/DYZ 23. Identity Property of Addition24. 90 = m/WBC 24. Identity Property of Addition

25. 90 + 90 = m/DYZ + m/WBC 25. Addition Property for Equality

26. 180 = m/DYZ + m/WBC 26. Substitution of Equals (90 + 90 = 180)

27. /DYZ and /WBC are supplementary 27. Definition of Supplementary Angles

STATEMENT REASON

A O

BC D

D E

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U

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CA– 4– 5 – 3 – 2 – 1 0 1

– 3– 4 – 2 – 1 0 1 2

NameUnit III, Part F, Lessons 3&4, Quiz Form B—Continued—


2. Solve the inequality 2y + 8 < – 6y, and graph the solution on a number line. Then answer the following:

a) Does the graph of the solution set of the inequality represent a ray?__________ Explain. __________

__________________________________________________________________________________

b) Write the inequality whose solution set is the complement of the solution set for 2y + 8 < –6y.

__________________________________________________________________________________

c) The union of the two solutions to the inequalities in part a) and part b) is a ________________ angle.straight

No the graph

does not have a definite endpoint

2y + 8 < –6y

2y + –2y + 8 < –6y + –2y

0 + 8 < –8y

8 < –8y

–1 > y or y < –1

1

–88 >

1

–8⋅

⋅ −8y

2y + 8 < –6y or 2y + 8 $ –6y

A O

BC D

D E

BA

C

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A

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A

B

CA D

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A B

1 2

X Y

DA

E

C

B

DA

CB

E

CA– 4– 5 – 3 – 2 – 1 0 1

– 3– 4 – 2 – 1 0 1 2


Name

Class Date Score

Quiz Form A

Unit III - Fundamental TermsPart G - Theorems About Segments and RaysLesson 1 - Theorem 13: “If two angles are complementary to the sameangle or congruent angles, then they are congruent to each other.”Lesson 2 - Theorem 14: “If two angles are supplementary to the sameangle or congruent angles, then they are congruent to each other.”Lesson 3 - Theorem 15: “If two lines intersect, then the vertical anglesformed are congruent.”

In the diagram, at the right, /AFB is a right angle. Name the figures described in exercises 1 through 6 below.

1. Another right angle _______________

2. Two complementary angles. _______________

3. Two congruent supplementary angles. _______________

4. Two non-congruent supplementary angles. _______________

5. Two acute vertical angles. _______________

6. Two obtuse vertical angles. _______________

In the diagram, at the right, OT bisects /SOU, m/UOV = 30 and m/YOW = 126. Find the measure of each angle.

7. m/VOW _______________

8. m/ZOY _______________

9. m/TOU _______________

10. m/ZOW _______________

11. m/UOS _______________

12. m/TOZ _______________

/AFE and /EFD

/BFA and /DFA/CFA and /EFA or/BFC and /DFC or

/BFE and /EFD

/BFC and /DFE

/CFD and /BFE

24O

30O

63O

156O

126O

87O

DFB

C

E

A

WS O

XY

Z

TU

V

DB

C

AF

E

x 2x+12P

F

E

B

C

D

A

14

2 3

B D

C

A

1

2

3 4 A BC

/AFD

DFB

C

E

A

WS O

XY

Z

TU

V

DB

C

AF

E

x 2x+12P

F

E

B

C

D

A

14

2 3

B D

C

A

1

2

3 4 A BC

NameUnit III, Part H, Lessons 1,2&3, Quiz Form B—Continued—


In the diagram at the right, p || q and t || r. Use this diagramto find m/12 in Exercises 5 through 12.

5. m/2 = 75O m/12 = _________ 6. m/3 = 110O m/12 = _________

7. m/16 = 80O m/12 = _________ 8. m/14 = 72O m/12 = _________

9. m/6 = 68O m/12 = _________ 10. m/5 = 104O m/12 = _________

11. m/12 + m/16 = 132O m/12 = ________ 12. m/8 + m/10 = 162O m/12 = _________

G F

C

D

B A

E

1 2 3 4 9 10

5 6 7 8

t1 t2

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C B

A

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70

(3x – 14)(2x – 10)

,1 ,2

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91

(5x – 24)

,1 ,2

(2x – 8),1

,2

(15x – 19) (7x – 3)

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9 10 13 14

11 12 15 16

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D G B

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A C

m

t

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m/2 = m/12 m/3 = m/1

m/12 + m/1 = 180

m/12 + 110 = 180

m/12 = 70

m/16 = m/12 m/14 = m/16

m/16 = m/12

m/14 = m/12

m/6 = m/16

m/16 = m/12

m/6 = m/12

m/5 + m/16 = 180

m/16 = m/12

m/5 + m/12 = 180

104 + m/12 = 180

m/12 = 76

m/12 = m/16

m/12 = 66

m/8 = m/16

m/16 = m/10

m/8 = m/10

m/10 = 81

m/10 = m/12

m/12 = 81

70O

80O 72O

68O 76O

66O 81O

75O

NameUnit III, Part H, Lessons 4,5,6&7, Quiz Form A—Continued—


13. Given: DG || AC and/ACE is a right angle as shown

Prove: /ABC and /FBG are complementary angle

G F

C

D

B A

E

1 2 3 4 9 10

5 6 7 8

t1 t2

,1

,2

C B

A

E D 1 2

3 4

P N

M

S

R Q

,1

,2

70

(3x – 14)(2x – 10)

,1 ,2

,1

,2

91

(5x – 24)

,1 ,2

(2x – 8),1

,2

(15x – 19) (7x – 3)

T

R

M

N P

Q

3 2

1 4

1 2 5 6 7 8

3 4

9 10 13 14

11 12 15 16

e f

c

d

p

q

4 1 3 2

12 9 11 10

8 5 7 6

16 13 15 14

t r

B

A E

C

D 1 2 3

B A

E D C

3 1

2 E

B

D

A

C

1 2 3 4

B

D

A

C 5 6 7 8

t r

D

A

B

E

C

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1 2

3

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(12x – 9) 135

E

D C

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1 3

2

C A B

F

E

D

D G B

F E

A C

m

t

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1 2

5 6

3 4

7 8

S

W

T

R

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Y X

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G H

A

B

C E

D

A B

D C

B

C E

G F

A

D

1

2

3

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t

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G B

J

D

C

A

I

H E

A

B

C F

D

4 3

2 1

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F

A

B

G

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t

t

t t

t

t

0

0

0

0 0

0

0

0

0 0

1. /ACE is a right angle 1. Given

2. CE AC 2. Definition of Perpendicular Lines.

3. DG || AC 3. Given

4. CE DG 4. Theorem 18 - If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

5. /DBC is a right angle 5. Definition of Perpendicular Lines

6. m/DBC = 90 6. Definition of Right Angle.

7. m/DBA + m/ABC = m/DBC 7. Postulate 7 (Protractor) - Angle-Addition Assumption

8. /DBA > /FBG 8. Theorem 15 - If two lines intersect, then the vertical angles formed are congruent

9. m/DBA = m/FBG 9. Definition of Congruent Angles

10. m/FBG + m/ABC = m/DBC 10. Substitution of Equality (9 into 7)

11. m/FBC + m/ABC = 90 11. Substitution of Equality (6 into 10)

12. /ABC and /FBC are complementary angles 12. Definition of Complementary Angles

STATEMENT REASON

>

>

NameUnit III, Test Form B—Continued—


2. Given: collinear points, P, Q, R, and SPQ > SR

Prove: PR > SQ

1. Points P, Q, R and S are collinear 1. Given

2. PQ > SR 2. Given

3. PQ = SR 3. Definition of Congruent Line Segments

4. QR = RQ 4. Reflexive Property of Equality

5. PQ + QR = SR + RC 5. Addition Property for Equality

6. PQ + QR = PR 6. Postulate 6 (Ruler) - Segment-Addition Assumption

7. SR + RC = SQ 7. Postulate 6 (Ruler) - Segment-Addition Assumption

8. PR = SQ 8. Substitution of Equality (statements 5, 6 and 7)

9. PR > SQ 9. Definition of Congruent Line Segments

STATEMENT REASON

1 23 4 9 10

5 6 7 8

t1 t2

,1

,2

BQ

N

E

C

D EA BF

D

A

C

E

B

1 2

3,1

,2

,3

M N Q

,

m

t

1 23 4

5 67 8

,

ts

m

1

2

3 45 6

7 89 10

,2

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1 23 4

75 6

8 E B

D T

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4

1

2

34

m

n

1 23 4

m5 67 8

,

t

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1 23 4

75 6

8

A T

R F

A B

C D

E F

1

2

ED C

B

A

F

E

G

A

CD

B

H

3

2

A

BDE

C

G F

1

p

q

2

431

RM S

,1 ,2

t A B C

D C D

P QN

A

M

B

P Q R S

T

T

Q,

m

(II-C-1,7)

Geometry: A Complete Course · are opposite rays, then the angles are supplementary.” For each of the following statements 1 through 10, write either true or false. 1. Two angles

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