Appendix A — Answer Key, Chapter 9: Angle Pair Relationships

SOL 8.MG.1 · Covers textbook Chapter 9 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 116 across the chapter. Reasoning answers show an acceptable response, not the only wording.

Every angle measure in this key has been checked against the relationship it comes from: each supplementary pair sums to exactly $180^\circ$, each complementary pair to exactly $90^\circ$, each vertical pair is equal, and every solved measure is positive and consistent with its diagram.

Naming note: an angle may be written with its three letters in either order, so $\angle ABC$ and $\angle CBA$ are both correct. Where a diagram names the angles with numbers, the number form is used here.

Lesson 9.1 — Angles, Adjacent Angles, and Adding Measures

Guided practice

1. $\angle ABC$, $\angle CBA$, and $\angle B$; the measure is $55^\circ$.
2. Vertex $B$; sides $\overrightarrow{BA}$ and $\overrightarrow{BC}$.
3. $\angle COB$ and $\angle BOA$. They share side $\overrightarrow{OB}$.
4. $m\angle COA = 40^\circ + 70^\circ = 110^\circ$
5. They share vertex $O$ and side $\overrightarrow{OC}$, but $\angle 4$ lies entirely inside $\angle 3$, so their interiors overlap. Adjacent angles must have non-overlapping interiors, so the second condition fails.
6. $55^\circ$ acute; $90^\circ$ right; $118^\circ$ obtuse; $180^\circ$ straight.

Independent practice

7. a) acute b) right c) obtuse d) acute e) straight
8. $\angle TSR$ and $\angle S$.
9. $m\angle FGJ = 52^\circ + 64^\circ = 116^\circ$
10. $m\angle ABC = 25^\circ + 61^\circ = 86^\circ$
11. $m\angle WYZ = 96^\circ - 43^\circ = 53^\circ$
12. Because $D$ is inside $\angle ABC$, the whole of $\angle ABD$ lies inside $\angle ABC$. The two angles do share vertex $B$ and side $\overrightarrow{BA}$, so the first condition holds, but their interiors overlap, so the second fails.
13. No. Adjacent angles must share a side as well as a vertex. Two angles meeting only at a point are like the two halves of a bow tie — next to each other in the picture, but with no common side.
14. The stylus and the lid are adjacent pieces of the $105^\circ$ angle, so their measures add to $105^\circ$: $105^\circ - 30^\circ = 75^\circ$. Check: $30 + 75 = 105$.
15. Yes. Adjacency says nothing about measure, only about position. Two adjacent angles of $100^\circ$ and $120^\circ$ are both obtuse; together they take up $220^\circ$ of the $360^\circ$ around the vertex, which is allowed. (What cannot happen is two obtuse angles forming a linear pair, since that would need a sum of $180^\circ$.)
16. Adjacency is about position, not size. The two angles share vertex $O$ and side $\overrightarrow{OB}$, and neither lies inside the other, so they are adjacent regardless of whether their measures are equal, close, or far apart.

Exit ticket 9.1

17. Vertex $Q$; sides $\overrightarrow{QP}$ and $\overrightarrow{QR}$.
18. $m\angle FGJ = 52^\circ + 64^\circ = 116^\circ$
19. (1) They share a vertex and a side. (2) Their interiors do not overlap.
20. Obtuse, because $179^\circ$ is greater than $90^\circ$ and less than $180^\circ$. It is close to straight but not straight, and "straight" means exactly $180^\circ$.

Lesson 9.2 — Complementary and Supplementary Angles

Guided practice

21. $90^\circ - 35^\circ = 55^\circ$
22. $180^\circ - 130^\circ = 50^\circ$
23. $90^\circ - 12^\circ = 78^\circ$
24. $180^\circ - 90^\circ = 90^\circ$
25. $\angle RSQ$ ($35^\circ$) and $\angle QSP$ ($55^\circ$); together they form the right angle $\angle PSR$. Check: $35 + 55 = 90$.
26. $\angle CBD$ ($130^\circ$) and $\angle DBA$ ($50^\circ$). Points $A$, $B$, and $C$ lie on one straight line, so $\angle ABC$ is a straight angle of $180^\circ$, and ray $\overrightarrow{BD}$ splits it into two pieces that must add to $180^\circ$. Check: $130 + 50 = 180$.
27. Complementary: yes, because $25 + 65 = 90$. Adjacent: no, because the angles have different vertices ($M$ and $V$) and share no side. Complementary is a statement about the sum; adjacent is a statement about position.
28. No. If one angle is greater than $90^\circ$, the other would have to have a negative measure to make the sum $90^\circ$, and no angle has a negative measure.

Independent practice

29. a) $85^\circ$ b) $49^\circ$ c) $21.5^\circ$ d) $1^\circ$
30. a) $165^\circ$ b) $90^\circ$ c) $63^\circ$ d) $1^\circ$
31. Let the smaller be $x$ degrees and the larger $3x$ degrees. Then $x + 3x = 90$, so $4x = 90$ and $x = 22.5$. The angles are $22.5^\circ$ and $67.5^\circ$. Check: $22.5 + 67.5 = 90$.
32. Let the smaller be $x$ degrees and the larger $5x$ degrees. Then $6x = 180$ and $x = 30$. The angles are $30^\circ$ and $150^\circ$. Check: $30 + 150 = 180$.
33. $m\angle B = 90^\circ - 54^\circ = 36^\circ$
34. $m\angle D = 180^\circ - 88^\circ = 92^\circ$
35. a) complementary, $47 + 43 = 90$ b) supplementary, $62 + 118 = 180$ c) complementary, $30 + 60 = 90$ d) neither, $105 + 85 = 190$
36. $x + x = 90$, so $x = 45$. Each angle measures $45^\circ$.
37. Two right angles add to $90 + 90 = 180$, which is exactly the supplementary condition. They can never be complementary, because a complementary pair sums to $90^\circ$ and a single right angle already uses all $90^\circ$, leaving $0^\circ$ for the other — and $0^\circ$ is not an angle.
38. The flagpole is vertical and the ground is level, so the string, the ground, and the pole form a right angle at the base: the two angles are complementary. $90^\circ - 68^\circ = 22^\circ$. Check: $68 + 22 = 90$.
39. The post is vertical and the floor is horizontal, so those two angles are complementary. $90^\circ - 55^\circ = 35^\circ$. Check: $55 + 35 = 90$.
40. The student found the complement instead of the supplement — $90 - 40 = 50$. A supplement is measured against $180^\circ$, so the supplement of $40^\circ$ is $180^\circ - 40^\circ = 140^\circ$. Check: $40 + 140 = 180$.

Exit ticket 9.2

41. $90^\circ - 73^\circ = 17^\circ$
42. $180^\circ - 73^\circ = 107^\circ$
43. $x + x = 180$, so $x = 90$. Each measures $90^\circ$, and each is a right angle.
44. An obtuse angle measures more than $90^\circ$ but less than $180^\circ$. Subtracting it from $180^\circ$ leaves a positive measure, so the supplement exists. Subtracting it from $90^\circ$ leaves a negative number, and no angle has a negative measure, so there is no complement.

Lesson 9.3 — Vertical Angles

Guided practice

45. $\angle 1$ and $\angle 3$; $\angle 2$ and $\angle 4$.
46. $\angle 1$ and $\angle 2$; $\angle 2$ and $\angle 3$; $\angle 3$ and $\angle 4$; $\angle 4$ and $\angle 1$.
47. $m\angle 2 = 180^\circ - 55^\circ = 125^\circ$; $m\angle 3 = 55^\circ$ (vertical to $\angle 1$); $m\angle 4 = 125^\circ$ (vertical to $\angle 2$). Check: $55 + 125 = 180$ for each neighboring pair.
48. All three measure $90^\circ$: the vertical partner equals $90^\circ$, and each neighbor is $180^\circ - 90^\circ = 90^\circ$. The lines are perpendicular.
49. $\angle 1$ and $\angle 2$ form a linear pair, so $m\angle 1 = 180^\circ - m\angle 2$. $\angle 2$ and $\angle 3$ also form a linear pair, so $m\angle 3 = 180^\circ - m\angle 2$. Both measures equal the same quantity $180^\circ - m\angle 2$, so $m\angle 1 = m\angle 3$.
50. No. Adjacent angles must share a side, and vertical angles share only the vertex — each side of one is the opposite ray of a side of the other, not the same ray.

Independent practice

51. $138^\circ$, $42^\circ$, $138^\circ$. The vertical partner is $42^\circ$; each neighbor is $180^\circ - 42^\circ = 138^\circ$.
52. $62^\circ$, $118^\circ$, $62^\circ$. Each neighbor is $180^\circ - 118^\circ = 62^\circ$.
53. Vertical angles are equal, so $3x = 51$ and $x = 17$. Check: $3(17) = 51$.
54. $7x - 4 = 5x + 12$, so $2x = 16$ and $x = 8$. Each angle measures $7(8) - 4 = 52^\circ$, and $5(8) + 12 = 52^\circ$ agrees.
55. Each measures $90^\circ$. If all four are equal and a neighboring pair sums to $180^\circ$, then each is $180 \div 2 = 90$. The lines are perpendicular.
56. Vertical angles are equal, so each is half of $130^\circ$: $m\angle 1 = m\angle 3 = 65^\circ$. Check: $65 + 65 = 130$.
57. Vertical angles are always equal. If an equal pair were also supplementary, then $x + x = 180$, giving $x = 90$. So the only supplementary vertical pair is a pair of right angles, which happens exactly when the lines are perpendicular.
58. $x = 35^\circ$, because $x$ is vertical to the $35^\circ$ angle and vertical angles are equal. $y = 180^\circ - 35^\circ = 145^\circ$, because $y$ and the $35^\circ$ angle form a linear pair and are therefore supplementary. Check: $35 + 145 = 180$, and the fourth angle is $145^\circ$, vertical to $y$.
59. Exactly two. The obtuse angle's vertical partner is equal to it, so that one is obtuse too. Each neighbor is the supplement of an angle greater than $90^\circ$, so each neighbor is less than $90^\circ$ — acute, not obtuse.
60. The student used the rule for a linear pair. Neighboring angles at an intersection are supplementary; vertical angles are equal. Counterexample: if $m\angle 1 = 55^\circ$ then its vertical partner is $55^\circ$, and $55 + 55 = 110$, not $180$.

Exit ticket 9.3

61. $107^\circ$, $73^\circ$, $107^\circ$. Each neighbor is $180^\circ - 73^\circ = 107^\circ$, and the vertical partner is $73^\circ$.
62. $5x = 85$, so $x = 17$. Check: $5(17) = 85$.
63. Vertical angles sit opposite each other at the intersection of two lines and share only the vertex; they are always equal. Adjacent angles sit beside each other, sharing a vertex and a side with non-overlapping interiors; their measures can be anything.
64. Because adjacency requires a shared side. In a vertical pair, the sides of one angle are the opposite rays of the sides of the other, so the two angles have no ray in common — only the vertex.

Lesson 9.4 — Solving for Unknown Angle Measures

Guided practice

65. The angles form a linear pair, so they are supplementary: $5x + (3x + 20) = 180$, so $8x = 160$ and $x = 20$. Measures: $5(20) = 100^\circ$ and $3(20) + 20 = 80^\circ$. Check: $100 + 80 = 180$.
66. Ray $\overrightarrow{SQ}$ splits a right angle, so the two pieces are complementary: $3x + (2x + 10) = 90$, so $5x = 80$ and $x = 16$. Measures: $3(16) = 48^\circ$ and $2(16) + 10 = 42^\circ$. Check: $48 + 42 = 90$.
67. The marked angles are vertical, so they are equal: $4x - 10 = 2x + 30$, giving $2x = 40$ and $x = 20$. Each marked angle measures $4(20) - 10 = 70^\circ$, and $2(20) + 30 = 70^\circ$ agrees. (The other two angles at $T$ each measure $110^\circ$.)
68. The three angles fill a straight angle: $40 + 2x + 70 = 180$, so $2x = 70$ and $x = 35$. The middle angle measures $2(35) = 70^\circ$. Check: $40 + 70 + 70 = 180$.
69. Complementary, because the post is square to the ground, so the ramp angle and the marked angle together fill a right angle. $90^\circ - 28^\circ = 62^\circ$. Check: $28 + 62 = 90$.
70. $2x + (4x + 12) = 180$, so $6x = 168$ and $x = 28$. Measures: $2(28) = 56^\circ$ and $4(28) + 12 = 124^\circ$. Check: $56 + 124 = 180$.
71. $(x + 15) + 2x = 90$, so $3x = 75$ and $x = 25$. Measures: $25 + 15 = 40^\circ$ and $2(25) = 50^\circ$. Check: $40 + 50 = 90$.
72. $6x + 5 = 8x - 21$, so $26 = 2x$ and $x = 13$. Each angle measures $6(13) + 5 = 83^\circ$, and $8(13) - 21 = 83^\circ$ agrees.

Independent practice

73. $(3x - 6) + (x + 10) = 180$, so $4x + 4 = 180$, $4x = 176$, and $x = 44$. Measures: $3(44) - 6 = 126^\circ$ and $44 + 10 = 54^\circ$. Check: $126 + 54 = 180$.
74. $5x + (x + 12) = 90$, so $6x = 78$ and $x = 13$. Measures: $5(13) = 65^\circ$ and $13 + 12 = 25^\circ$. Check: $65 + 25 = 90$.
75. $9x - 14 = 7x + 2$, so $2x = 16$ and $x = 8$. Each angle measures $9(8) - 14 = 58^\circ$, and $7(8) + 2 = 58^\circ$ agrees.
76. $55 + 3x + 65 = 180$, so $3x = 60$ and $x = 20$. The middle angle measures $3(20) = 60^\circ$. Check: $55 + 60 + 65 = 180$.
77. Let the smaller be $x$ degrees, so the larger is $x + 18$ degrees. Then $x + (x + 18) = 90$, so $2x = 72$ and $x = 36$. Measures: $36^\circ$ and $54^\circ$. Check: $36 + 54 = 90$, and $54 - 36 = 18$.
78. Let one be $x$ degrees, so the other is $3x - 24$ degrees. Then $x + (3x - 24) = 180$, so $4x = 204$ and $x = 51$. Measures: $51^\circ$ and $3(51) - 24 = 129^\circ$. Check: $51 + 129 = 180$, and $129$ is indeed $24$ less than $3(51) = 153$.
79. Let the angle be $x$ degrees. Its supplement is $180 - x$ and its complement is $90 - x$, so $180 - x = 4(90 - x)$. Expanding gives $180 - x = 360 - 4x$, so $3x = 180$ and $x = 60$. The angle is $60^\circ$. Check: the supplement is $120^\circ$, the complement is $30^\circ$, and $4(30) = 120$.
80. The post is vertical and the ground is level, so the ramp angle and the angle between ramp and post are complementary: $90^\circ - 22^\circ = 68^\circ$. Check: $22 + 68 = 90$.
81. The two named angles are neighbors at the crossing, so they form a linear pair: $4x + (5x + 45) = 180$, giving $9x = 135$ and $x = 15$. The angles are $4(15) = 60^\circ$ and $5(15) + 45 = 120^\circ$, and their vertical partners repeat those measures. All four: $60^\circ$, $120^\circ$, $60^\circ$, $120^\circ$. Check: $60 + 120 = 180$.
82. The three angles lie along a straight edge, so they are supplementary as a group: $30 + 2x + (x + 15) = 180$, so $3x + 45 = 180$, $3x = 135$, and $x = 45$. Measures: $30^\circ$, $2(45) = 90^\circ$, and $45 + 15 = 60^\circ$. Check: $30 + 90 + 60 = 180$.
83. The student used the supplementary sum, $180$, for a complementary pair. Complementary means the sum is $90$, so the equation is $3x + (x + 20) = 90$, giving $4x = 70$ and $x = 17.5$. Measures: $3(17.5) = 52.5^\circ$ and $17.5 + 20 = 37.5^\circ$. Check: $52.5 + 37.5 = 90$.
84. A straight line makes a straight angle of $180^\circ$ at every point on it. So as soon as one or more rays split that line at a point, the pieces are adjacent angles that must add to $180^\circ$ — a known total. The equation comes from the total, not from any single marked measure, which is why the line itself never needs a label.

Exit ticket 9.4

85. $(4x + 10) + (2x + 20) = 180$, so $6x + 30 = 180$, $6x = 150$, and $x = 25$. Measures: $4(25) + 10 = 110^\circ$ and $2(25) + 20 = 70^\circ$. Check: $110 + 70 = 180$.
86. $7x + 2x = 90$, so $9x = 90$ and $x = 10$. Measures: $70^\circ$ and $20^\circ$. Check: $70 + 20 = 90$.
87. $5x - 12 = 3x + 8$, so $2x = 20$ and $x = 10$. Each angle measures $5(10) - 12 = 38^\circ$, and $3(10) + 8 = 38^\circ$ agrees.
88. Complementary: set the sum of the two expressions equal to $90$. Supplementary: set the sum equal to $180$. Vertical: set the two expressions equal to each other.

Chapter 9 Review

Part A — Identifying and describing angle pair relationships (8.MG.1a)

89. Vertical: two angles opposite each other where two lines cross, sharing only the vertex; they are always equal. Adjacent: two angles that share a vertex and a side and do not overlap. Supplementary: two angles whose measures add to $180^\circ$. Complementary: two angles whose measures add to $90^\circ$.
90. Adjacent: $\angle EBD$ and $\angle DBC$ (they share side $\overrightarrow{BD}$). Complementary: $\angle EBD$ ($35^\circ$) and $\angle DBC$ ($55^\circ$), since $35 + 55 = 90$. Supplementary: $\angle ABC$ ($90^\circ$) and $\angle CBE$ ($90^\circ$), since $90 + 90 = 180$. Other correct answers exist — for example $\angle ABD$ ($145^\circ$) and $\angle DBE$ ($35^\circ$) is another supplementary pair, since $145 + 35 = 180$.
91. $a = 55^\circ$, from the complementary relationship: $\overrightarrow{BC} \perp$ line $AE$, so $\angle EBC = 90^\circ$, and $35 + a = 90$. $b = 90^\circ$, because $\overrightarrow{BC}$ is perpendicular to line $AE$ — equivalently from the supplementary relationship along the line, $35 + 55 + b = 180$.
92. The $118^\circ$ angle and $\angle 3$; $\angle 2$ and $\angle 4$.
93. $m\angle 2 = 180^\circ - 118^\circ = 62^\circ$ (linear pair with the $118^\circ$ angle). $m\angle 3 = 118^\circ$ (vertical to the $118^\circ$ angle). $m\angle 4 = 62^\circ$ (vertical to $\angle 2$). Check: $118 + 62 = 180$ for every neighboring pair, and $118 + 62 + 118 + 62 = 360$.
94. $\angle 2$ and $\angle 4$. Those are the two angles that share a side with $\angle 1$; $\angle 3$ is its vertical partner and shares only the vertex.
95. Vertical: never adjacent. Supplementary: sometimes adjacent (a linear pair is; two separate angles of $100^\circ$ and $80^\circ$ are not). Complementary: sometimes adjacent (the two pieces of a split right angle are; $25^\circ$ and $65^\circ$ drawn apart are not).
96. False. Supplementary is a condition on the sum only. Two angles of $120^\circ$ and $60^\circ$ drawn on opposite sides of the page share no vertex and no side, so they are not adjacent, yet $120 + 60 = 180$ makes them supplementary.
97. Adjacency constrains position, not measure, so two adjacent angles may add to anything. For example, adjacent angles of $40^\circ$ and $30^\circ$ share a vertex and a side, but $40 + 30 = 70$, which is neither $90$ nor $180$.
98. The left panel shows adjacent angles, because $\angle 1$ and $\angle 2$ share vertex $O$ and side $\overrightarrow{OB}$ with neither angle inside the other. The right panel is not adjacent, because $\angle 4$ lies inside $\angle 3$, so their interiors overlap.

Part B — Solving for unknown angle measures (8.MG.1b)

99. Complement: $90^\circ - 26^\circ = 64^\circ$. Supplement: $180^\circ - 26^\circ = 154^\circ$.
100. $180^\circ - 149^\circ = 31^\circ$
101. $(2x + 5) + (3x - 20) = 180$, so $5x - 15 = 180$, $5x = 195$, and $x = 39$. Measures: $2(39) + 5 = 83^\circ$ and $3(39) - 20 = 97^\circ$. Check: $83 + 97 = 180$.
102. $(4x + 2) + (x - 7) = 90$, so $5x - 5 = 90$, $5x = 95$, and $x = 19$. Measures: $4(19) + 2 = 78^\circ$ and $19 - 7 = 12^\circ$. Check: $78 + 12 = 90$, and both measures are positive.
103. $10x - 3 = 7x + 18$, so $3x = 21$ and $x = 7$. Each angle measures $10(7) - 3 = 67^\circ$, and $7(7) + 18 = 67^\circ$ agrees.
104. $x + 62 + (2x + 4) = 180$, so $3x + 66 = 180$, $3x = 114$, and $x = 38$. Measures: $38^\circ$, $62^\circ$, and $2(38) + 4 = 80^\circ$. Check: $38 + 62 + 80 = 180$.
105. $147^\circ$, $33^\circ$, $147^\circ$. Each neighbor is $180^\circ - 33^\circ = 147^\circ$, and the vertical partner is $33^\circ$.
106. Let the measures be $2k$ and $3k$ degrees. Then $2k + 3k = 90$, so $5k = 90$ and $k = 18$. Measures: $36^\circ$ and $54^\circ$. Check: $36 + 54 = 90$, and $36 : 54$ reduces to $2 : 3$.
107. Let the smaller be $x$ degrees, so the larger is $x + 46$ degrees. Then $x + (x + 46) = 180$, so $2x = 134$ and $x = 67$. Measures: $67^\circ$ and $113^\circ$. Check: $67 + 113 = 180$, and $113 - 67 = 46$.
108. Let the angle be $x$ degrees; its supplement is $180 - x$. Then $x = 3(180 - x)$, so $x = 540 - 3x$, $4x = 540$, and $x = 135$. The angle is $135^\circ$ and its supplement is $45^\circ$. Check: $135 + 45 = 180$, and $3(45) = 135$.
109. $25 + 35 + 3x + 30 = 180$, so $3x + 90 = 180$, $3x = 90$, and $x = 30$. The unknown angle measures $3(30) = 90^\circ$. Check: $25 + 35 + 90 + 30 = 180$.
110. $x + 40 = 3x$, so $40 = 2x$ and $x = 20$. Each angle measures $20 + 40 = 60^\circ$, and $3(20) = 60^\circ$ agrees.

Part C — Mixed application and reasoning

111. The post is still square to the ground, so the relationship is still complementary: $90^\circ - 34^\circ = 56^\circ$. Check: $34 + 56 = 90$. Raising the ramp by $6^\circ$ lowered the marked angle by the same $6^\circ$, from $62^\circ$ to $56^\circ$, which is what a fixed total of $90^\circ$ forces.
112. $x = 48^\circ$, because $x$ is vertical to the marked angle and vertical angles are equal. $y = 180^\circ - 48^\circ = 132^\circ$, because $y$ and the marked angle form a linear pair. Check: $48 + 132 = 180$.
113. Vertical angles are equal, so $6x - 9 = 4x + 15$, giving $2x = 24$ and $x = 12$. Those two angles each measure $6(12) - 9 = 63^\circ$, and $4(12) + 15 = 63^\circ$ agrees. The remaining two angles are each $180^\circ - 63^\circ = 117^\circ$. All four: $63^\circ$, $117^\circ$, $63^\circ$, $117^\circ$. Check: $63 + 117 = 180$, and the four sum to $360^\circ$.
114. Let the angle be $x$ degrees. Then $180 - x = 3(90 - x)$, so $180 - x = 270 - 3x$, $2x = 90$, and $x = 45$. The angle is $45^\circ$. Check: the supplement is $135^\circ$, the complement is $45^\circ$, and $3(45) = 135$.
115. Vertical angles are equal, not supplementary. The student was thinking of a linear pair — two neighboring angles at the intersection, whose outer sides form a straight line, which do add to $180^\circ$. A correct statement: "$\angle 1$ and $\angle 2$ are vertical angles, so $m\angle 1 = m\angle 2$." The student's equation only happens to be true in the special case where both angles are $90^\circ$.
116. Look for one of three features. A straight line through the vertex means the angles along it are supplementary, so set their sum equal to $180$. A right angle, marked with a small square or supplied by a physical vertical-meets-level situation, means the pieces inside it are complementary, so set their sum equal to $90$. Two lines crossing means the opposite angles are vertical, so set the two expressions equal to each other. Then solve, substitute back for the measures rather than stopping at $x$, and check that every measure is positive and matches the picture.

Workbook-only items

Page 2, fill in the blanks. The shared endpoint is the vertex. The two rays are the sides. Three names: $\angle ABC$, $\angle CBA$, $\angle B$. $m\angle ABC = \mathbf{55^\circ}$.

Page 2, classify table. greater than $0^\circ$ and less than $90^\circ$ — acute; exactly $90^\circ$ — right; greater than $90^\circ$ and less than $180^\circ$ — obtuse; exactly $180^\circ$ — straight.

Page 3, two conditions. (1) They share a vertex and share a side. (2) Their interiors do not overlap.

Page 3, item 4 frame. $m\angle COA = \mathbf{40^\circ} + \mathbf{70^\circ} = \mathbf{110^\circ}$.

Page 6, definitions. Complementary angles add to $90^\circ$; supplementary angles add to $180^\circ$. For a complement subtract from $90^\circ$; for a supplement subtract from $180^\circ$. A complementary pair is made of two acute angles, so an obtuse angle has no complement.

Page 10, fill in the blanks. Vertical angles sit opposite each other and share only the vertex. Vertical angles are never adjacent. Each pair of neighbors is a linear pair, so each neighboring pair adds to $180^\circ$.

Page 11, item 49 frame. $m\angle 1 = 180^\circ - \mathbf{m\angle 2}$ and $m\angle 3 = 180^\circ - \mathbf{m\angle 2}$, so the two measures are equal, because both equal the same quantity.

Page 13, the routine. Find the relationship: a straight line means supplementary, a right angle means complementary, two lines crossing means vertical. Write the equation. Solve, then substitute back and check.

Page 13, item 65 frame. Equation: $5x + (3x + 20) = 180$. $x = \mathbf{20}$. Measures $\mathbf{100^\circ}$ and $\mathbf{80^\circ}$. Check: $100 + 80 = 180$.

Page 13, item 66 frame. Equation: $3x + (2x + 10) = 90$. $x = \mathbf{16}$. Measures $\mathbf{48^\circ}$ and $\mathbf{42^\circ}$. Check: $48 + 42 = 90$.

Page 14, item 67 frame. Equation: $4x - 10 = 2x + 30$. $x = \mathbf{20}$. Each marked angle $\mathbf{70^\circ}$.

Page 14, item 68 frame. Equation: $40 + 2x + 70 = 180$. $x = \mathbf{35}$. Middle angle $\mathbf{70^\circ}$.

Page 17, item 79 frame. Equation: $180 - x = 4(90 - x)$. Angle $60^\circ$. Check: supplement $120^\circ$, complement $30^\circ$, and $4 \times \mathbf{30} = \mathbf{120}$.

Page 17, item 83 frame. Correct: $x = \mathbf{17.5}$, measures $\mathbf{52.5^\circ}$ and $\mathbf{37.5^\circ}$.

Page 17, item 88 table. complementary — sum $= 90$; supplementary — sum $= 180$; vertical — the two expressions are equal.

Page 19, item 95 table. vertical — never; supplementary — sometimes; complementary — sometimes.

Page 22, item 114 frame. Equation: $180 - x = 3(90 - x)$. Angle $45^\circ$. Check: supplement $135^\circ$, complement $45^\circ$.

Page 22, item 116 frame. Straight line means supplementary, so the sum is $180$. Right angle means complementary, so the sum is $90$. Crossing lines mean vertical, so the expressions are equal.