Geometry Workbook — Chapter 13: Circles: Central and Inscribed Angles

SOL G.PC.3 (b, c) · Companion to Textbook Chapter 13

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 126.

PAGE 1 — Chapter opener

Chapter 13 · Circles: Central and Inscribed Angles

Standard G.PC.3 (b, c)

In this chapter you will:

• Name the parts of a circle and the three kinds of arc
• Use arcs around a circle total $360°$ to find a missing arc
• Use a central angle equals its arc
• Use an inscribed angle is half its arc — and justify it
• Get three famous results out of that one rule
• Decide, before computing, where the vertex is

Words to know: circle · centre · radius · chord · diameter · arc · minor arc · major arc · semicircle · arc measure · central angle · inscribed angle · subtend · inscribed quadrilateral

Conventions: arcs are in degrees only (length is Chapter 14). Minor arc = two letters, major arc = three. Arcs add, and total $360°$. Congruent arcs need one circle — equal measure is not equal size.

PAGE 2 — The whole chapter on one page

Find the Vertex First

Everything else in this chapter is the second row again. Same arc, same endpoints, different vertex — and the answers differ by a factor of two.

Before every problem:

• Where is the vertex — at the centre, or on the circle?
• Am I going arc → angle (halve) or angle → arc (double)?

PAGE 3 — The parts of a circle

13.1 Anatomy

FIGURE: fig1-parts-of-a-circle.png (full width)

Fill in the table.

1. Name the segment from the centre to the circle. ____________

2. What makes a chord a diameter? ____________________

3. Diameter $=$ ______ $\times$ radius

PAGE 4 — Three kinds of arc

Minor, Major, Semicircle

FIGURE: fig2-naming-arcs.png (full width)

4. The minor arc is ____________ and measures ______

5. The major arc is ____________ and measures ______

   Why does it need three letters? ____________________

$$\text{minor} + \text{major} = 360°$$

PAGE 5 — Arcs add and total 360°

Around the Whole Circle

FIGURE: fig4-arcs-total-360.png (full width)

6. The four arcs are ______ ______ ______ ______ , totalling ______

Most "find the missing arc" questions need no theorem at all — add what you have and subtract from $360$.

PAGE 6 — Practice · parts and arcs

Practice

7. A chord through the centre is called a ____________

8. Radius $9$. Diameter $=$ ______

9. Diameter $26$. Radius $=$ ______

10. $\widehat{AB} = 74°$. Minor or major? ____________

11. $\widehat{AB} = 74°$. The major arc measures ______

12. $\widehat{CD} = 128°$. The major arc measures ______

13. Three arcs are $70°$, $140°$, $60°$. The fourth is ______

14. Three arcs are $45°$, $135°$, $90°$. The fourth is ______

15. Three arcs are $100°$, $60°$, $85°$. The fourth is ______

16. $\widehat{PQ} = 65°$, $\widehat{QR} = 85°$. $\widehat{PR} =$ ______

17. $\widehat{AB} = 40°$, $\widehat{BC} = 75°$, $\widehat{CD} = 95°$. $\widehat{AD} =$ ______

18. A semicircle measures ______ An arc is a semicircle when ____________________

PAGE 7 — Think it through · 13.1

Think It Through

19. Application. A round pizza is cut into $8$ equal slices. Give the arc measure of one slice's crust. ______

20. Error analysis. A student writes $\widehat{AB} = 250°$. What is wrong with the notation, and how do you fix it?

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21. Reasoning. Why does an arc's measure tell you nothing about its length?

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22. Reasoning. Why must the minor and major arcs on the same two points total $360°$?

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Exit ticket 13.1

23. $\widehat{MN} = 96°$. The major arc measures ______

24. Three arcs are $110°$, $85°$, $75°$. The fourth is ______

PAGE 8 — The angle is the arc

13.2 Central Angles

FIGURE: fig3-central-equals-arc.png (full width)

25. The angle is ______ and the arc is ______ They are related by ____________________

26. Why is this a definition rather than a theorem?

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$$m\angle AOB = \widehat{AB}$$

PAGE 9 — Equal angles, equal arcs, equal chords

The Three-Way Chain

FIGURE: fig5-congruent-central-angles.png (full width)

27. Both angles are ______ and both arcs are ______

28. State the chain: congruent ____________ $\Leftrightarrow$ congruent ____________ $\Leftrightarrow$ congruent ____________

29. Why does the chain need the words in one circle?

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30. $n$ points equally spaced → each arc is ____________

PAGE 10 — Practice · central angles

Practice

$O$ is the centre throughout.

31. $m\angle AOB = 75°$. $\widehat{AB} =$ ______

32. $m\angle COD = 128°$. $\widehat{CD} =$ ______

33. $\widehat{EF} = 64°$. $m\angle EOF =$ ______

34. $\widehat{GH} = 95°$. $m\angle GOH =$ ______

35. $m\angle AOB = 110°$. Major arc $\widehat{AXB} =$ ______

36. $m\angle COD = 145°$. The major arc measures ______

37. Six points equally spaced. Central angle between neighbours: ______

38. Ten points equally spaced. Central angle: ______

39. Nine points equally spaced. Central angle: ______

40. Two congruent central angles; one cuts a $68°$ arc. The other arc is ______

PAGE 11 — Practice · central angles with algebra

Solve for the Variable

41. Three central angles fill a circle: $4y$, $5y$, $3y$.

    Equation: ____________________ $y =$ ______ Arcs: ______ ______ ______

42. Four central angles fill a circle: $2x$, $3x$, $4x$, $6x$.

    $x =$ ______ Arcs: ______ ______ ______ ______

43. $\widehat{AB} = 50°$ and $\widehat{BC} = 110°$. $m\angle AOC =$ ______

PAGE 12 — Think it through · 13.2

Think It Through

44. Application. A Ferris wheel has $8$ cars equally spaced. Central angle between neighbours: ______

    Arc a car travels between two stops: ______

45. Application. A pie chart shows one category as a $54°$ sector. Its share of the whole, as a fraction in lowest terms: ______

46. Error analysis. A student says a $70°$ arc on a bicycle wheel is congruent to a $70°$ arc on a merry-go-round. Correct them.

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47. Reasoning. Why can a central angle never exceed $360°$? What does an angle of exactly $180°$ cut off?

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48. Reasoning. Two chords in one circle are congruent. What follows about their arcs and central angles?

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Exit ticket 13.2

49. $m\angle AOB = 84°$. $\widehat{AB} =$ ______ Major arc: ______

50. Five points equally spaced. Central angle: ______

PAGE 13 — Half the arc

13.3 Inscribed Angles

FIGURE: fig6-inscribed-is-half.png (full width)

51. The arc is ______ and the inscribed angle is ______

52. The central angle on the same arc is ______ The ratio central : inscribed is ______

53. An inscribed angle has its vertex ____________ and its sides are ____________

$$m\angle ACB = \tfrac12 \widehat{AB}$$

Going backwards, double. Inscribed $37°$ → arc $74°$.

PAGE 14 — Why it is half

The Proof

FIGURE: fig7-why-half.png (full width)

54. Why is $\triangle OVP$ isosceles? ____________________

55. Which theorem makes the angle at $O$ equal to $2x$? ____________________

56. Why is the angle at $O$ the same as the arc? ____________________

The chain: two radii → isosceles → base angles both $x$ → exterior angle at $O$ is $2x$ → that is the arc → arc is twice the inscribed angle.

PAGE 15 — Practice · inscribed angles

Practice

Give the inscribed angle for each arc.

57. $110°$ → ______

58. $86°$ → ______

59. $74°$ → ______

60. $208°$ → ______

61. $96°$ → ______

Give the arc for each inscribed angle.

62. $34°$ → ______

63. $63°$ → ______

64. $27°$ → ______

65. $48°$ → ______

66. $70°$ → ______

PAGE 16 — Practice · both angles, and algebra

Practice

67. Arc $128°$. Central ______ Inscribed ______

68. Arc $250°$. Central ______ Inscribed ______

69. Inscribed $55°$. Arc ______ Central ______

70. Inscribed $(3x + 5)°$, arc $(8x - 10)°$.

    Equation: ____________________ $x =$ ______ Angle ______ Arc ______

71. Inscribed $(2x)°$, arc $(5x - 30)°$.

    $x =$ ______ Angle ______ Arc ______

PAGE 17 — Think it through · 13.3

Think It Through

72. Application. A camera on a circular gallery wall views a painting whose ends cut off a $74°$ arc. The angle the camera must cover: ______

73. Error analysis. Told an inscribed angle is $40°$, a student answers "the arc is $20°$." Identify the error.

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74. Error analysis. A student sees a vertex on the circle and writes "angle $=$ arc." Which rule did they use, and which did they need?

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75. Reasoning. Why can an inscribed angle never reach $180°$?

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76. Reasoning. How does the diameter case prove the general case?

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Exit ticket 13.3

77. Inscribed angle on a $96°$ arc: ______

78. Inscribed angle $63°$ → arc ______

PAGE 18 — Same arc

13.4 Three Consequences

FIGURE: fig8-same-arc.png (full width)

79. The arc is ______ and both angles are ______ Why equal? ____________________

80. What happens if the vertex slides along the major arc? ____________________

PAGE 19 — The semicircle

A Diameter Makes a Right Angle

FIGURE: fig9-angle-in-a-semicircle.png (full width)

81. Arc ______ Angle ______ Arithmetic: ______

82. What must be true of $\overline{MN}$? ____________________

Worth recognising instantly: a diameter and a point on the circle make a right angle.

PAGE 20 — The inscribed quadrilateral

Opposite Angles

FIGURE: fig10-inscribed-quadrilateral.png (full width)

83. The two pairs of opposite angles: ______ $+$ ______ $=$ ______ and ______ $+$ ______ $=$ ______

84. Why do adjacent angles have no such rule?

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PAGE 21 — Practice · the three consequences

Practice

85. Two inscribed angles on the same arc; one is $34°$. The other: ______

86. Same arc; one is $52°$. The other ______ The arc ______

87. $\overline{MN}$ a diameter, $V$ on the circle. $m\angle MVN =$ ______

88. In that figure, $m\angle VMN = 35°$. $m\angle VNM =$ ______

89. In that figure, $m\angle VMN = 52°$. $m\angle VNM =$ ______

90. $PQRS$ inscribed, $m\angle P = 112°$. $m\angle R =$ ______

91. $PQRS$ inscribed, $m\angle Q = 95°$. $m\angle S =$ ______

92. $PQRS$ inscribed, $m\angle P = 105°$, $m\angle Q = 100°$. $m\angle R =$ ______ $m\angle S =$ ______

PAGE 22 — Practice · quadrilaterals from arcs

Practice

93. Opposite angles $(2x + 10)°$ and $(3x + 20)°$.

    $x =$ ______ Angles ______ and ______

94. Arcs in order $70°$, $90°$, $110°$, $90°$. All four angles: ______ ______ ______ ______

95. Arcs in order $40°$, $140°$, $90°$, $90°$. All four angles: ______ ______ ______ ______

96. An inscribed angle is $90°$. Its arc ______ The chord joining its endpoints is a ____________

PAGE 23 — Think it through · 13.4

Think It Through

97. Application. A carpenter needs a right angle but has only a straightedge and a round tabletop. What do you draw, and why does it work?

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98. Application. A window has all four corners on a circular frame. Two opposite corners measure $88°$ and $92°$. Possible? ______ Explain.

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99. Error analysis. A student says two inscribed angles are equal "because their vertices are close together." Give the correct reason, and say when two inscribed angles are not equal.

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100. Error analysis. A student adds two adjacent angles of an inscribed quadrilateral and expects $180°$. Correct them.

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101. Reasoning. Show that all three of this lesson's results follow from the halving rule.

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102. Reasoning. A triangle is inscribed with one side a diameter. Why must it be right-angled, and where is the right angle?

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Exit ticket 13.4

103. $\overline{AB}$ a diameter, $C$ on the circle. $m\angle ACB =$ ______

104. Inscribed quadrilateral, $m\angle A = 43°$. The opposite angle: ______

PAGE 24 — The one decision

13.5 Central or Inscribed?

FIGURE: fig11-central-vs-inscribed.png (full width)

105. The two angles are ______ and ______ The one thing that differs: ____________________

106. State the two-line rule.

     At the centre → ____________________

     On the circle → ____________________

FIGURE: fig12-rules-table.png (full width)

107. Which two rows are worth memorising? ____________________ Why not the rest?

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108. The semicircle row is a special case of ____________________

PAGE 25 — In context

Two Uses

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

109. Central angle between neighbouring cars: ______ Arithmetic: ______

110. Explain the carpenter's right angle. ____________________

PAGE 26 — Practice · which rule?

Central or Inscribed — Then Solve

111. Vertex at the centre, arc $128°$. ____________ Angle ______

112. Vertex on the circle, arc $128°$. ____________ Angle ______

113. Vertex on the circle, arc $180°$. ____________ Angle ______

114. Vertex at the centre, arc $210°$. ____________ Angle ______

115. Vertex on the circle, arc $74°$. ____________ Angle ______

116. $\widehat{AB} = 50°$, $\widehat{BC} = 110°$. Inscribed angle on $\widehat{AC}$: ______

117. Inscribed $63°$. Central on the same arc: ______

118. Central $96°$. Inscribed on the same arc: ______

119. Three arcs are $100°$, $60°$, $85°$. The fourth ______ Inscribed angle on it ______

PAGE 27 — Think it through · 13.5

Think It Through

120. Application. A round window has $12$ equally spaced spokes from the centre. Central angle ______ Inscribed angle on one of those arcs ______

121. Application. Two markers on a dish rim cut off a $146°$ arc. A sensor at the centre and one on the rim both aim at both markers.

     Centre sensor: ______ Rim sensor: ______

122. Error analysis. A student halves a central angle. What did they confuse, and what is the correct rule?

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123. Reasoning. How can you tell from a figure alone which rule applies?

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124. Reasoning. Why can an inscribed angle and a central angle on the same arc never be equal?

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Exit ticket 13.5

125. Vertex on the circle, arc $208°$. Angle ______

126. The one question to ask before using either rule: ____________________

PAGE 28 — Blank circles

Your Turn

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

A checklist for every problem in this chapter:

• Where is the vertex? At the centre, or on the circle?
• At the centre → angle $=$ arc. On the circle → angle $=$ half arc.
• Going arc → angle, halve. Going angle → arc, double.
• Missing an arc? Add what you have and subtract from $360$.
• Is a diameter in the picture? Then some arc is $180°$ and some inscribed angle is $90°$.

PAGE 29 — Chapter review

Chapter 13 Review

Review 1 (G.PC.3b). A circle is divided by four radii into arcs of $60°$, $120°$, $80°$, and one more.

• Give the fourth arc, and the fact you used.
• Give the four central angles.
• Two adjacent arcs measure $60°$ and $120°$. Give the central angle across both, then name the arc that is left — carefully.
• What would change if the radius doubled? What would not?

Review 2 (G.PC.3c). $\overline{AB}$ is a diameter of circle $O$; $C$ is on the circle with $\widehat{AC} = 74°$.

• $m\angle ACB =$ ______ Reason: ____________________
• $\widehat{CB} =$ ______
• $m\angle ABC =$ ______ Reason: ____________________
• Check all three against the triangle sum, and name the rule used at each step.

PAGE 30 — Chapter review, continued

Review 3 (G.PC.3 b, c). $PQRS$ is inscribed in a circle; the arcs $\widehat{PQ}$, $\widehat{QR}$, $\widehat{RS}$, $\widehat{SP}$ measure $50°$, $110°$, $100°$, $100°$.

• Check that the four arcs are possible. ______
• Give all four angles.

• Verify both pairs of opposite angles: ______ $+$ ______ $=$ ______ and ______ $+$ ______ $=$ ______
• Give $m\angle QOR$ ______ and say why it differs from $m\angle QPR$ ______

PAGE 31 — Vocabulary check

Words to Know

circle · centre · radius · chord · diameter · arc · minor arc · major arc · semicircle · arc measure · central angle · inscribed angle · subtend · inscribed quadrilateral

The two that carry the chapter:

• Central angle — vertex at the centre. The angle is the arc. (This is the definition of arc measure.)
• Inscribed angle — vertex on the circle. The angle is half the arc.

And the three results that are just the second one again:

• Two angles on the same arc → congruent (both are half the same number).
• An angle in a semicircle → $90°$ (half of $180$).
• Opposite angles of an inscribed quadrilateral → supplementary (the two arcs make $360$).

Answer keys for every item are in Appendix A.