Geometry Workbook — Chapter 12: Angles of Convex Polygons

SOL G.PC.2 (a, b, c) · Companion to Textbook Chapter 12

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 112.

PAGE 1 — Chapter opener

Chapter 12 · Angles of Convex Polygons

Standard G.PC.2 (a, b, c)

In this chapter you will:

• Say what convex means, and why the chapter needs it
• Justify the interior sum $(n-2)180°$ by cutting a polygon into triangles
• Justify why the exterior angles of any convex polygon total $360°$
• Find each interior and exterior angle of a regular polygon
• Run both relationships backwards to find $n$
• Spot an angle that describes no regular polygon at all

Words to know: polygon · convex · non-convex · regular · interior angle · exterior angle · supplementary · interior angle sum · $n$-gon · diagonal · vertex · tile

Conventions: convex only. Regular = equal sides and equal angles. At every vertex, interior + exterior = 180°. And $n$ is a whole number — a fraction means no such polygon.

PAGE 2 — The two pairs of formulas

The Whole Chapter on One Page

The right-hand column needs the word regular. An irregular hexagon has an interior sum of $720°$ — but it has no single "each interior angle" at all. Reading across the wrong row is the most common error in this chapter.

Before every problem, ask two questions:

• Is it a sum or is it each?
• If it is each — am I told the polygon is regular?

PAGE 3 — What convex rules out

12.1 Convex

FIGURE: fig1-convex-and-not.png (full width)

1. Define convex in your own words. ________________________________________

2. The marked angle on the right measures ______ , which proves ____________________

3. (from the next figure) How many triangles? ______ From how many diagonals? ______

PAGE 4 — The justification

Cut It Into Triangles

FIGURE: fig2-cut-into-triangles.png (full width)

4. Give the sum and the arithmetic that produced it. ____________________

5. Why are the triangles' angles exactly the polygon's angles, with nothing left over?

   ---

The rule, rebuilt rather than recalled: from one vertex you draw $n - 3$ diagonals, which make $n - 2$ triangles, each worth $180°$.

$$\text{sum of interior angles} = (n-2) \cdot 180°$$

PAGE 5 — The sum table

The Sum Depends on $n$ Alone

FIGURE: fig3-interior-sum-table.png (full width)

Fill in the blanks.

6. By how much does the sum grow when one side is added? ______ Why? ____________________

Nothing in this table mentions regular. A lopsided hexagon and a perfect one both total $720°$.

PAGE 6 — Practice · sums and sides

Practice

Find the interior angle sum.

7. Quadrilateral. ______

8. Hexagon. ______

9. Nonagon. ______

10. $15$-gon. ______

11. $20$-gon. ______

Find the number of sides.

12. Sum $900°$. ______

13. Sum $1260°$. ______

14. Sum $2340°$. ______

15. Sum $3240°$. ______

PAGE 7 — Practice · missing angles and algebra

Practice

Find the missing angle.

16. Pentagon; four angles are $100°$, $120°$, $95°$, $130°$. ______

17. Quadrilateral; three angles are $85°$, $95°$, $100°$. ______

18. Hexagon; five angles are $110°$, $130°$, $120°$, $140°$, $100°$. ______

19. Heptagon; six angles are $120°$, $130°$, $140°$, $125°$, $135°$, $110°$. ______

FIGURE: fig12-angle-sum-with-algebra.png (half width, right)

20. Quadrilateral angles in the ratio $3:4:5:6$.

    Equation: ____________________ $x =$ ______ Angles: ______ ______ ______ ______

21. Quadrilateral angles in the ratio $2:3:4:6$.

    $x =$ ______ Angles: ______ ______ ______ ______

PAGE 8 — Think it through · 12.1

Think It Through

22. Application. A garden bed is a convex hexagon. Five corners measure $115°$, $125°$, $130°$, $105°$, $120°$. Give the sixth, and say how you know it without measuring.

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23. Error analysis. A student computes a hexagon's interior sum as $6 \times 180° = 1080°$. Find the error and give the correct sum.

    ---

24. Reasoning. Why does the interior angle sum not depend on whether the polygon is regular?

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Exit ticket 12.1

25. Interior sum of a convex octagon: ______

26. Angles total $1440°$. Sides: ______

27. Quadrilateral; three angles $70°$, $110°$, $95°$. Fourth: ______

28. Justify $(n-2)180°$ in one sentence, without stating the formula.

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PAGE 9 — What an exterior angle is

12.2 The Extension Is the Point

FIGURE: fig4-what-an-exterior-angle-is.png (full width)

29. What is the dashed ray, and why is it needed?

    ---

30. The two angle measures are ______ and ______ , and their sum is ______

31. Why are an interior and an exterior angle at the same vertex always supplementary?

    ---

interior + exterior = 180° is the hinge of this chapter. It turns every interior question into an exterior one, and back.

PAGE 10 — Why the exterior sum is 360°

Walk Around Once

FIGURE: fig5-walking-the-boundary.png (full width)

32. At each corner you turn through ____________________

33. Why is one trip around the boundary $360°$? ____________________

$$\text{sum of exterior angles} = 360°, \quad \textbf{whatever } n \textbf{ is}$$

Notice what is not in that statement: $n$. A triangle's three exterior angles total $360°$; a $20$-gon's twenty exterior angles total $360°$. More angles, each smaller, same total.

PAGE 11 — It does not need regularity

Nothing Equal, Totals Unchanged

FIGURE: fig6-irregular-still-360.png (full width)

34. The two totals are ______ and ______

    What is this figure here to prove? ____________________

PAGE 12 — Practice · exterior angles

Practice

35. Exterior sum of a triangle. ______

36. Exterior sum of a $12$-gon. ______

37. Exterior sum of a $30$-gon. ______

38. Interior $115°$ → exterior ______

39. Interior $140°$ → exterior ______

40. Exterior $40°$ → interior ______

41. Exterior $24°$ → interior ______

Find the missing exterior angle.

42. Pentagon; four are $80°$, $65°$, $70°$, $75°$. ______

43. Hexagon; five are $70°$, $55°$, $60°$, $65°$, $50°$. ______

44. Quadrilateral; three are $95°$, $85°$, $100°$. ______

45. A convex $9$-gon. Exterior sum ______ Interior sum ______

46. A convex polygon's interior angles total $1080°$. Exterior sum: ______

PAGE 13 — Think it through · 12.2

Think It Through

47. Application. A robot drives once around a convex flower bed and returns facing its starting direction. Give the total of all its turns, and say why it does not depend on the number of corners.

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48. Application. A quadrilateral gate has interior angles $95°$, $85°$, $100°$, $80°$. Give all four exterior angles and check the total.

    ______ ______ ______ ______ Total: ______

49. Error analysis. A student writes: "a hexagon has $6$ sides, so its exterior angles total $6 \times 60° = 360°$; a pentagon has $5$, so $5 \times 60° = 300°$." Find the error.

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50. Error analysis. A student says an irregular pentagon's exterior angles cannot total $360°$ because they are all different. Correct them.

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51. Reasoning. Why does adding a side increase the interior sum but leave the exterior sum unchanged?

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52. Reasoning. Show that the exterior sum being $360°$ forces the interior sum to be $180n - 360$, and check this equals $(n-2)180$.

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Exit ticket 12.2

53. Exterior sum of any convex polygon: ______

54. Interior $128°$ → exterior ______

55. Hexagon; five exteriors $60°$, $70°$, $50°$, $80°$, $45°$. Sixth: ______

56. Justify the $360°$ in one sentence. ____________________

PAGE 14 — Divide, but only when regular

12.3 One Angle at a Time

FIGURE: fig7-regular-pentagon.png (full width)

57. Each interior angle: ______ ÷ ______ = ______

58. Each exterior angle: ______ ÷ ______ = ______

59. Check them against each other: ______ + ______ = ______ What does the check confirm?

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$$\text{each interior} = \frac{(n-2)180°}{n} \qquad \text{each exterior} = \frac{360°}{n}$$

The exterior formula is easier. $360 \div n$ needs no sum first — and the interior angle is then one subtraction away.

PAGE 15 — As n grows

Flattening Toward a Circle

FIGURE: fig8-three-regular-polygons.png (full width)

Fill in.

60. What happens to each angle as $n$ grows, and why?

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PAGE 16 — The reference table, and the trap

Read the Right Column

FIGURE: fig9-regular-polygon-table.png (full width)

61. Which columns need the polygon to be regular? ____________________

    Which do not? ____________________

FIGURE: fig13-polygons-in-context.png (full width)

62. Why do three hexagons close around a point and pentagons not?

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PAGE 17 — Practice · each angle

Practice

Each interior angle of a regular polygon with:

63. $6$ sides ______

64. $9$ sides ______

65. $10$ sides ______

66. $15$ sides ______

67. $20$ sides ______

Each exterior angle of a regular polygon with:

68. $8$ sides ______

69. $12$ sides ______

70. $18$ sides ______

71. $24$ sides ______

72. $36$ sides ______

73. Each exterior is $45°$. Each interior: ______

74. Each interior is $160°$. Each exterior: ______

75. Regular $12$-gon: interior sum ______ each interior ______

76. An irregular octagon has interior sum $1080°$. Each interior angle: ____________________

PAGE 18 — Think it through · 12.3

Think It Through

77. Application. A stop sign is a regular octagon. One corner: ______

    A saw cuts along the extension of one edge. What angle is it set to? ______ Why?

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78. Application. Show that squares tile a floor and regular pentagons do not.

    Squares: ____________________ Pentagons: ____________________

79. Error analysis. A student finds each interior angle of a regular pentagon as $360 \div 5 = 72°$. Find the error and give the correct value.

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80. Reasoning. Why is each interior angle of a regular polygon always less than $180°$, however large $n$ is?

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Exit ticket 12.3

81. Each interior of a regular hexagon: ______

82. Each exterior of a regular $20$-gon: ______

83. Each interior of that $20$-gon: ______

84. An irregular pentagon has interior sum $540°$. Each interior angle: ____________________

PAGE 19 — Backwards from an exterior angle

12.4 One Division

FIGURE: fig10-backwards-from-exterior.png (full width)

85. The one-line rule: $n =$ ____________________

86. Why is the exterior angle the easier door in?

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PAGE 20 — Backwards from an interior angle

Subtract First, Then Divide

FIGURE: fig11-backwards-from-interior.png (full width)

87. The two steps, in order:

    1. ---

    2. ---

88. In the shaded row, what happens and what is the correct answer?

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89. Why must $n$ be a whole number? ____________________

90. Why does going straight at the interior angle give the same answer?

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$n$ must be a whole number. "About $10$" is not an answer. "No such polygon" is.

PAGE 21 — Practice · find n

Practice

From each exterior angle of a regular polygon:

91. $72°$ → $n =$ ______

92. $40°$ → $n =$ ______

93. $30°$ → $n =$ ______

94. $20°$ → $n =$ ______

95. $12°$ → $n =$ ______

From each interior angle of a regular polygon:

96. $108°$ → $e =$ ____ , $n =$ ____

97. $135°$ → $e =$ ____ , $n =$ ____

98. $144°$ → $e =$ ____ , $n =$ ____

99. $156°$ → $e =$ ____ , $n =$ ____

100. $165°$ → $e =$ ____ , $n =$ ____

Does a regular polygon exist? Give $n$ if it does.

101. Each exterior $7°$. ____________________

102. Each interior $145°$. ____________________

103. Each interior $100°$. ____________________

104. Each interior $170°$. ____________________

PAGE 22 — Think it through · 12.4

Think It Through

105. A convex polygon's interior angles total $1980°$. $n =$ ______

106. A regular polygon has each interior angle three times each exterior angle. Find $n$.

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107. Application. A machinist cuts regular plates whose corners are $150°$. How many sides? ______

     What angle does the cutting guide make with the edge? ______

108. Error analysis. Told each interior angle is $145°$, a student answers "about a $10$-gon." Why is that wrong rather than merely imprecise?

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Exit ticket 12.4

109. Each exterior $24°$. $n =$ ______

110. Each interior $150°$. $n =$ ______

111. Each interior $130°$. $n =$ ____________________

112. The two-step route from an interior angle to $n$: ____________________

PAGE 23 — Blank frames

Your Turn

FIGURE: fig14-blank-polygon-frames.png (full width)

A checklist for every problem in this chapter:

• Count the sides. That is $n$.
• Is the question a sum or each?
• If each — is the polygon regular? If not, the question may have no answer.
• Going backwards? Get to the exterior angle first, then divide $360$ by it.
• Is $n$ a whole number? If not, the polygon does not exist.

PAGE 24 — Chapter review

Chapter 12 Review

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

• Give the interior angle sum, and justify it by describing the triangle cut — not by citing the formula.
• Give the exterior angle sum, and justify it in one sentence.
• Six interior angles are $120°$, $130°$, $140°$, $125°$, $135°$, $110°$. Find the seventh.
• Give the exterior angle at that seventh vertex, and name the relationship you used.

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

• Give the interior angle sum.
• Give each interior and each exterior angle, and check them against each other.
• Which of those three answers survives if the polygon is not regular? Which does not?
• A second regular polygon has each exterior angle exactly twice this one's. Give its number of sides and each interior angle.

PAGE 25 — Chapter review, continued

Review 3 (G.PC.2a). A designer is choosing a regular polygon for a paving stone.

• The first candidate has each interior angle $140°$. Give $n$.
• The second has each interior angle $155°$. Give $n$, or say why there is none.
• The stones must tile with no gaps, so a whole number of corners must meet at each point. Show that the first candidate fails this test even though it exists.
• Name every regular polygon that does tile a floor by itself, and show the arithmetic for each.

PAGE 26 — Vocabulary check

Words to Know

polygon · convex · non-convex · regular · interior angle · exterior angle · supplementary · interior angle sum · $n$-gon · diagonal · vertex · tile

The three that carry the chapter:

• Convex — every interior angle under $180°$. Everything here assumes it.
• Regular — equal sides and equal angles. Only then is there such a thing as "each angle."
• Supplementary — interior $+$ exterior $= 180°$ at every vertex. This is what lets you move between the two halves of the chapter.

And the two justifications, which are worth more than the formulas:

• $n$ sides cut into $n-2$ triangles → interior sum $(n-2)180°$
• one walk around the boundary is one full turn → exterior sum $360°$

Answer keys for every item are in Appendix A.