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Written by: Larry E. Collins
Geometry:A Complete Course
(with Trigonometry)
Module C – Instructor's Guidewith Detailed Solutions for
Progress Tests
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
© 2006 VideoTextInteractive Geometry: A Complete Course 25
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
D
B
F
H
C
G
x
y A B C
DF
E
P
C
D
BA
DB
E
CA
12 4
3
z
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
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
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
(x)O
(3x)O13
9
> >
> >
(4x – 1)O
© 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
© 2006 VideoTextInteractive Geometry: A Complete Course 41
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—
© 2006 VideoTextInteractive Geometry: A Complete Course 43
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
T
NM 31 2
4Q
X
Z
Y
A
C
B DW
21
,
,1
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
STATEMENT REASON
© 2006 VideoTextInteractive Geometry: A Complete Course 51
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
,
,1
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—
© 2006 VideoTextInteractive Geometry: A Complete Course 55
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
BA
C
U
V
T
W
X
Y
R
T
NM 31 2
4Q
X
Z
Y
A
C
B DW
21
,
,1
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 B—Continued—
© 2006 VideoTextInteractive Geometry: A Complete Course58
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
U
V
T
W
X
Y
R
T
NM 31 2
4Q
X
Z
Y
A
C
B DW
21
,
,1
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
© 2006 VideoTextInteractive Geometry: A Complete Course 63
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—
© 2006 VideoTextInteractive Geometry: A Complete Course78
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
,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
F
1 2
3
,1
,2
(12x – 9) 135
E
D C
B A
1 3
2
C A B
F
E
D
D G B
F E
A C
m
t
,
1 2
5 6
3 4
7 8
S
W
T
R
Z
Y X
Q P
N M E F
G H
A
B
C E
D
A B
D C
B
C E
G F
A
D
1
2
3
,1
,2
,3
t
E F
G B
J
D
C
A
I
H E
A
B
C F
D
4 3
2 1
E
F
A
B
G
H D
C
t
t
t t
t
t
0
0
0
0 0
0
0
0
0 0
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—
© 2006 VideoTextInteractive Geometry: A Complete Course 87
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
F
1 2
3
,1
,2
(12x – 9) 135
E
D C
B A
1 3
2
C A B
F
E
D
D G B
F E
A C
m
t
,
1 2
5 6
3 4
7 8
S
W
T
R
Z
Y X
Q P
N M E F
G H
A
B
C E
D
A B
D C
B
C E
G F
A
D
1
2
3
,1
,2
,3
t
E F
G B
J
D
C
A
I
H E
A
B
C F
D
4 3
2 1
E
F
A
B
G
H D
C
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—
© 2006 VideoTextInteractive Geometry: A Complete Course114
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
,3
,1
1 23 4
75 6
8 E B
D T
12 3
4
1
2
34
m
n
1 23 4
m5 67 8
,
t
,2
,3
,1
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)