Let’s go through each problem step by step, explaining the concepts and providing the correct answers.
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Part 1: Name the arc made by the given angle
> In a circle, a
central angle (vertex at the center) intercepts an
arc between its two endpoints. The arc is named using the two endpoints (and sometimes a point in between if it's a major arc).
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[1]
Angle: ∠HOG (vertex at O, points H and G on circle)
→ Arc:
Arc HG (minor arc, since angle is less than 180°)
✔ Answer: Arc HG
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[2]
Angle: ∠HOJ (vertex at O, points H and J on circle)
→ Arc:
Arc HJ
✔ Answer: Arc HJ
---
[3]
Angle: ∠HOK (vertex at O, points H and K on circle)
→ Arc:
Arc HK
✔ Answer: Arc HK
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Part 2: Name the central angle of the given arc
> The
central angle corresponding to an arc has its vertex at the center and endpoints at the arc’s endpoints.
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[4]
Arc: ML (from M to L, passing through the bottom)
→ Central angle:
∠MOL
✔ Answer: ∠MOL
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[5]
Arc: LP (from L to P, minor arc)
→ Central angle:
∠LOP
✔ Answer: ∠LOP
---
[6]
Arc: PQ (from P to Q)
→ Central angle:
∠POQ
✔ Answer: ∠POQ
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Part 3: If the angle is given, name the corresponding arc it makes
> Again, the central angle defines the arc between its two endpoints.
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[7]
Angle: ∠POR → Points P and R
→ Arc:
Arc PR
✔ Answer: Arc PR
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[8]
Angle: ∠AOB → Points A and B
→ Arc:
Arc AB
✔ Answer: Arc AB
---
[9]
Angle: ∠WOX → Points W and X
→ Arc:
Arc WX
✔ Answer: Arc WX
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Part 4: Find the measure of the arc indicated
>
Key Concept: The
measure of an arc (in degrees) is equal to the measure of its
central angle.
> *Note: The radius or length given is irrelevant for arc degree measure — only the central angle matters.*
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[10]
Central angle: ∠AOB = 120°
→ Arc AB =
120°
✔ Answer: 120°
*(The “5 cm” is the radius — distractor for arc length, but question asks for arc measure in degrees.)*
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[11]
Central angle: ∠AOX = 144°
→ Arc AX =
144°
✔ Answer: 144°
*(“8 in” is radius — again, not needed for degree measure.)*
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[12]
Central angle: ∠XOY = 65°
→ Arc XY =
65°
✔ Answer: 65°
*(“4 ft” is radius — irrelevant for arc degree measure.)*
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##
✔ Final Answers:
[1] Arc HG
[2] Arc HJ
[3] Arc HK
[4] ∠MOL
[5] ∠LOP
[6] ∠POQ
[7] Arc PR
[8] Arc AB
[9] Arc WX
[10] 120°
[11] 144°
[12] 65°
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## 📌 Summary of Key Concepts:
-
Central Angle: Angle with vertex at the center of the circle.
-
Arc Measure: Equal to the measure of its central angle (in degrees).
-
Naming Arcs: Use endpoints (e.g., Arc AB). For major arcs, include a third point (e.g., Arc ACB), but these problems use minor arcs.
-
Radius/Length Given? Only matters if you’re calculating
arc length (using formula: `Arc Length = (θ/360) × 2πr`). But here, we only need
degree measure, so ignore those numbers.
Let me know if you want to calculate arc lengths too!
Parent Tip: Review the logic above to help your child master the concept of degrees of a circle worksheet.
