Appendix A — Answer Key, Chapter 12: Angles of Convex Polygons

SOL G.PC.2 (a, b, c) · Covers textbook Chapter 12 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 112 across the chapter.

Conventions used in every answer below. Everything here assumes the polygon is convex — every interior angle less than $180°$. Regular means all sides congruent and all angles congruent. At every vertex, interior + exterior = $180°$, and that is the hinge most of these answers turn on. And $n$ is a whole number: an angle that produces a fraction describes no polygon at all, so "no such polygon" is a complete answer and a rounded $n$ is a wrong one.

The right-hand column is the one that needs regular. Asking for "each interior angle" of an irregular polygon is not a hard question — it is a question with no answer, and two items below say exactly that.

Reference values used throughout.

Lesson 12.1 — Convex Polygons and the Interior Angle Sum

Guided practice

1. Every interior angle is less than $180°$ — no vertex is pushed inward. (Equivalently: every diagonal stays inside the figure.)
2. $270°$. It is more than $180°$, so that figure is not convex and nothing in this chapter applies to it.
3. 5 triangles, from 4 diagonals — from one vertex you can reach every vertex but itself and its two neighbours, so $n - 3 = 4$.
4. $5 \times 180° =$ $900°$.
5. Because the diagonals only divide angles that were already there. Every angle of every triangle is either a whole angle of the polygon or one piece of one, each polygon angle is fully accounted for, and nothing outside the polygon is included. So the triangles' angles add up to exactly the polygon's angles — no more, no less.
6. $180°$, because adding one side adds exactly one more triangle to the cut.

Independent practice

7. $(4-2)180° =$ $360°$.
8. $(6-2)180° =$ $720°$.
9. $(9-2)180° =$ $1260°$.
10. $(15-2)180° =$ $2340°$.
11. $(20-2)180° =$ $3240°$.
12. $(n-2)180 = 900 \Rightarrow n-2 = 5 \Rightarrow n =$ $7$.
13. $n - 2 = 7 \Rightarrow n =$ $9$.
14. $n - 2 = 13 \Rightarrow n =$ $15$.
15. $n - 2 = 18 \Rightarrow n =$ $20$.
16. Sum $540°$; $100 + 120 + 95 + 130 = 445$, so the fifth is $540 - 445 =$ $95°$.
17. Sum $360°$; $85 + 95 + 100 = 280$, so the fourth is $360 - 280 =$ $80°$.
18. Sum $720°$; the five given total $600$, so the sixth is $720 - 600 =$ $120°$.
19. Sum $900°$; the six given total $760$, so the seventh is $900 - 760 =$ $140°$.
20. $3x + 4x + 5x + 6x = 360 \Rightarrow 18x = 360 \Rightarrow x = 20$, so the angles are $60°$, $80°$, $100°$, $120°$. (Check: they total $360$.)
21. $2x + 3x + 4x + 6x = 360 \Rightarrow 15x = 360 \Rightarrow x = 24$, so the angles are $48°$, $72°$, $96°$, $144°$. (Check: they total $360$.)
22. A hexagon's angles total $(6-2)180° = 720°$. The five given total $115 + 125 + 130 + 105 + 120 = 595$, so the sixth is $720 - 595 =$ $125°$. It is known without measuring because the sum is fixed by the number of sides alone — the shape of the bed cannot change it.
23. They multiplied by $n$ instead of $n - 2$. A hexagon cuts into 4 triangles, not $6$, so the sum is $(6-2)180° =$ $720°$.
24. Because the triangle cut works on any convex polygon, and the number of triangles depends only on the number of vertices, not on the sizes of the angles. Making a polygon lopsided moves angle measure from one vertex to another without changing the total.

Exit ticket 12.1

25. $(8-2)180° =$ $1080°$.
26. $(n-2)180 = 1440 \Rightarrow n - 2 = 8 \Rightarrow n =$ $10$.
27. $360 - (70 + 110 + 95) = 360 - 275 =$ $85°$.
28. Cutting from one vertex splits an $n$-gon into $n-2$ triangles whose angles are exactly the polygon's, and each triangle contributes $180°$.

Lesson 12.2 — Exterior Angles

Guided practice

29. It is the extension of the previous side. It is needed because an exterior angle is measured between a side and that extension — without the extension there is no second ray to measure to.
30. $108°$ and $72°$; their sum is $180°$.
31. Because a side and its own extension form a straight line, which is $180°$, and the interior and exterior angles at that vertex together fill exactly that straight angle.
32. One exterior angle.
33. Because you finish facing the direction you started, and that is one complete turn — $360°$.
34. $540°$ and $360°$. It is there to prove that neither total needs the polygon to be regular: all five angles differ and both totals hold anyway.

Independent practice

35. $360°$.
36. $360°$.
37. $360°$. (The number of sides never enters.)
38. $180 - 115 =$ $65°$.
39. $180 - 140 =$ $40°$.
40. $180 - 40 =$ $140°$.
41. $180 - 24 =$ $156°$.
42. $80 + 65 + 70 + 75 = 290$, so the fifth is $360 - 290 =$ $70°$.
43. $70 + 55 + 60 + 65 + 50 = 300$, so the sixth is $360 - 300 =$ $60°$.
44. $95 + 85 + 100 = 280$, so the fourth is $360 - 280 =$ $80°$.
45. Exterior sum $360°$; interior sum $(9-2)180° =$ $1260°$.
46. $360°$ — the interior sum tells you $n = 8$, but the exterior sum would be $360°$ whatever $n$ turned out to be.
47. $360°$ — one full turn. It does not depend on the number of corners because more corners means more turns, each of them smaller, and the trip is one revolution either way.
48. The exterior angles are $180 - 95 = 85°$, $180 - 85 = 95°$, $180 - 100 = 80°$, and $180 - 80 = 100°$. Check: $85 + 95 + 80 + 100 =$ $360°$. ✓
49. The exterior sum is $360°$ for every convex polygon — it does not depend on $n$. The hexagon answer is right by accident (a regular hexagon does have $60°$ exterior angles, and $6 \times 60 = 360$), but the pentagon reasoning is wrong twice: a regular pentagon's exterior angles are $72°$, not $60°$, and in any case the total is $360°$, not $300°$.
50. The total does not require the angles to be equal. The irregular pentagon in this lesson has five different exterior angles — $45°$, $71.57°$, $95.91°$, $72.78°$, and $74.74°$ — and they total exactly $360°$. Equality of the angles is what you need to find one of them, not to find their sum.
51. Adding a side adds one triangle to the interior cut, so the interior sum gains $180°$. But the walk around the boundary is still one full revolution however many corners it has, so the exterior sum is unchanged at $360°$.
52. At each vertex the interior and exterior angles total $180°$, and there are $n$ vertices, so all of them together total $180n$. The exterior angles account for $360$ of that, so the interior angles total $180n - 360$. And $180n - 360 = 180(n-2) = (n-2)180$ — the same formula. ✓

Exit ticket 12.2

53. $360°$.
54. $180 - 128 =$ $52°$.
55. $60 + 70 + 50 + 80 + 45 = 305$, so the sixth is $360 - 305 =$ $55°$.
56. Walking once around the boundary turns you through each exterior angle exactly once and leaves you facing your starting direction — one full turn, $360°$.

Lesson 12.3 — Regular Polygons, One Angle at a Time

Guided practice

57. $540° \div 5 =$ $108°$ — the interior sum divided by the number of angles, which is legitimate only because they are all equal.
58. $360° \div 5 =$ $72°$.
59. $108 + 72 = 180$. ✓ It confirms that the two are the supplementary pair at a vertex, so an arithmetic slip in either one would show up here.
60. The interior angle grows toward $180°$ and the exterior shrinks toward $0°$, because each exterior angle is $360/n$, which shrinks as $n$ grows. The polygon is flattening out toward a circle.
61. Each interior and each exterior need the polygon to be regular. The number of sides and the interior sum do not.
62. Three hexagons: $3 \times 120° = 360°$, exactly filling the space around the point. Pentagons: $360 \div 108 = \tfrac{10}{3}$, not a whole number, so no number of them closes the gap.

Independent practice

63. $720 \div 6 =$ $120°$.
64. $1260 \div 9 =$ $140°$.
65. $1440 \div 10 =$ $144°$.
66. $2340 \div 15 =$ $156°$.
67. $3240 \div 20 =$ $162°$.
68. $360 \div 8 =$ $45°$.
69. $360 \div 12 =$ $30°$.
70. $360 \div 18 =$ $20°$.
71. $360 \div 24 =$ $15°$.
72. $360 \div 36 =$ $10°$.
73. $180 - 45 =$ $135°$.
74. $180 - 160 =$ $20°$.
75. Sum $(12-2)180° =$ $1800°$; each interior $1800 \div 12 =$ $150°$.
76. You cannot. The interior sum is $1080°$, but irregular means the angles need not be equal, so there is no single "each interior angle" to give. Only the total is determined.
77. Each corner is $1080 \div 8 =$ $135°$. The saw follows the extension of an edge, so it is set to the exterior angle: $180 - 135 =$ $45°$.
78. Squares work: each interior angle is $90°$, and $360 \div 90 = 4$ — four squares meet exactly around a point. Regular pentagons do not: each interior angle is $108°$, and $360 \div 108 = \tfrac{10}{3}$, not a whole number, so three leave a gap and four overlap.
79. $360 \div n$ is the exterior angle, not the interior one. Each interior angle of a regular pentagon is $540 \div 5 =$ $108°$ — their $72°$ is the exterior angle, which is the supplement of the right answer.
80. Because each interior angle is $180 - \dfrac{360}{n}$, and $\dfrac{360}{n}$ is positive for every finite $n$. So the interior angle is always strictly less than $180°$ — it approaches $180°$ as $n$ grows but never reaches it, which is also exactly what "convex" requires.

Exit ticket 12.3

81. $720 \div 6 =$ $120°$.
82. $360 \div 20 =$ $18°$.
83. $180 - 18 =$ $162°$.
84. You cannot. The sum is $540°$, but an irregular pentagon has no single "each interior angle."

Lesson 12.4 — Working Backwards to $n$

Guided practice

85. $n = 360 \div e$, where $e$ is each exterior angle.
86. Because the exterior total is always $360°$, with no $n$ on the other side of the equation — one division finishes it. The interior formula has $n$ in both the numerator and the denominator, so it needs rearranging first.
87. (1) $e = 180 - i$. (2) $n = 360 \div e$.
88. $180 - 145 = 35°$, and $360 \div 35 \approx 10.3$, which is not a whole number — so no such regular polygon exists.
89. Because $n$ counts sides, and a polygon cannot have a fractional number of them.
90. Because they are the same equation rearranged. $\dfrac{(n-2)180}{n} = i$ is equivalent to $180 - \dfrac{360}{n} = i$, that is $\dfrac{360}{n} = 180 - i = e$, which is exactly the two-step route.

Independent practice

91. $360 \div 72 =$ $5$.
92. $360 \div 40 =$ $9$.
93. $360 \div 30 =$ $12$.
94. $360 \div 20 =$ $18$.
95. $360 \div 12 =$ $30$.
96. $e = 72°$, so $n =$ $5$.
97. $e = 45°$, so $n =$ $8$.
98. $e = 36°$, so $n =$ $10$.
99. $e = 24°$, so $n =$ $15$.
100. $e = 15°$, so $n =$ $24$.
101. $360 \div 7 \approx 51.4$ — not a whole number, so no such regular polygon exists.
102. $e = 35°$ and $360 \div 35 \approx 10.3$ — no such regular polygon exists.
103. $e = 80°$ and $360 \div 80 = 4.5$ — no such regular polygon exists.
104. $e = 10°$ and $360 \div 10 = 36$ — yes, a regular $36$-gon.
105. $(n-2)180 = 1980 \Rightarrow n - 2 = 11 \Rightarrow n =$ $13$.
106. Let $e$ be each exterior angle; then $i = 3e$, and $i + e = 180$, so $4e = 180$ and $e = 45°$. Then $n = 360 \div 45 =$ $8$. (Check: a regular octagon has interior $135°$ and exterior $45°$, and $135 = 3 \times 45$. ✓)
107. $i = 150° \Rightarrow e = 30° \Rightarrow n = 360 \div 30 =$ $12$ sides. The guide follows the extension of the edge, so it is set to the exterior angle, $30°$.
108. Because $n$ counts sides, and a polygon cannot have $10.3$ of them — the figure described does not exist, so rounding produces a different polygon rather than an approximate answer to the question asked. A regular $10$-gon has interior angles of $144°$, not $145°$. The correct answer is "no such regular polygon."

Exit ticket 12.4

109. $360 \div 24 =$ $15$.
110. $e = 30°$, so $n = 360 \div 30 =$ $12$.
111. $e = 50°$ and $360 \div 50 = 7.2$ — not a whole number, so no such regular polygon exists.
112. $e = 180 - i$, then $n = 360 \div e$.

Chapter 12 Review — answers

Review 1 (G.PC.2b). A convex heptagon.

• $900°$. Justification: pick one vertex and draw all four diagonals from it. They cut the heptagon into five triangles, and the triangles' angles are exactly the heptagon's angles — nothing added, nothing left over. Five triangles at $180°$ each gives $900°$.
• $360°$. Justification: walking once around the boundary turns you through each exterior angle exactly once and returns you to your starting direction, which is one complete turn.
• $120 + 130 + 140 + 125 + 135 + 110 = 760$, so the seventh is $900 - 760 =$ $140°$.
• $180 - 140 =$ $40°$, using interior + exterior $= 180°$ at a single vertex.

Review 2 (G.PC.2c). A regular $18$-gon.

• $(18-2)180° =$ $2880°$.
• Each interior $= 2880 \div 18 =$ $160°$; each exterior $= 360 \div 18 =$ $20°$. Check: $160 + 20 = 180$. ✓
• The interior sum, $2880°$, would still be available — it holds for every convex $18$-gon. Each interior and each exterior would not: without equal angles there is no single "each."
• Each exterior $= 40°$, so $n = 360 \div 40 =$ $9$ sides, and each interior angle is $180 - 40 =$ $140°$.

Review 3 (G.PC.2a).

• $i = 140° \Rightarrow e = 40° \Rightarrow n = 360 \div 40 =$ $9$. A regular nonagon.

• $i = 155° \Rightarrow e = 25° \Rightarrow 360 \div 25 = 14.4$ — not a whole number, so there is no such regular polygon.

• The nonagon exists but does not tile. For stones to meet with no gap, a whole number of corners must fill $360°$ — and $360 \div 140 = \tfrac{18}{7}$, which is not a whole number. Two nonagons leave a gap of $80°$ and three overlap. Existing and tiling are different tests, and passing the first says nothing about the second.

• Exactly three regular polygons tile a floor alone:

  Polygon	Each interior	$360 \div$ interior
  equilateral triangle	$60°$	$6$ ✓
  square	$90°$	$4$ ✓
  regular hexagon	$120°$	$3$ ✓

  Every other regular polygon fails: the pentagon gives $\tfrac{10}{3}$, the octagon $\tfrac{8}{3}$, the $12$-gon $\tfrac{12}{5}$. Beyond the hexagon every interior angle is more than $120°$ and less than $180°$, so two corners are not enough to fill $360°$ and three are too many — which is why the list stops.