Educational math worksheet for learning about angles and arcs in circles.

Sure! Let’s solve each problem step by step, explaining the concepts as we go.

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Key Concepts:

- Arc: A portion of the circumference of a circle.
- Central Angle: An angle whose vertex is at the center of the circle. The measure of the central angle equals the measure of its intercepted arc.
- Naming Arcs: Typically named using the endpoints (e.g., arc AB), and if it’s a major arc or needs clarification, a third point may be used (e.g., arc ACB).
- Measure of an Arc: Equal to the measure of its central angle (in degrees).

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## Part 1: Name the arc made by the given angle

[1]

Angle: ∠HOG (vertex O, rays OH and OG)
→ The arc between H and G, not passing through D → Arc HG
*(Since no other points are on the minor arc, we can just name it with two letters.)*

✔ Answer: Arc HG

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[2]

Angle: ∠HOI (rays OH and OI)
→ The arc between H and I → Arc HI

✔ Answer: Arc HI

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[3]

Angle: ∠HOK (rays OH and OK)
→ But notice: there’s a point N on the opposite side. Since the angle opens from H to K going past N, this is likely the major arc. So we name it with three points: Arc HNK or Arc HK passing through N.

But typically, if the angle is drawn as shown (smaller angle), it might be intended as the minor arc. However, in this diagram, since the angle appears to go from H to K via the longer path (passing N), and the angle symbol is drawn that way — it’s ambiguous. But in most worksheets, if the angle is drawn opening wide like that, it’s the major arc.

Wait — actually, looking again: the angle ∠HOK is drawn with the small wedge near the center, so it’s likely the minor arc HK.

But point N is directly opposite? Hmm. Actually, the angle shown is probably meant to be the minor arc HK, since it's less than 180°.

✔ Answer: Arc HK

*(Note: If it were the reflex angle, it would be labeled differently. We assume minor arc unless specified.)*

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## Part 2: Name the central angle of the given arc

[4]

Arc: ML (from M to L)
→ Central angle formed by radii OM and OL → ∠MOL

✔ Answer: ∠MOL

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[5]

Arc: LP (from L to P)
→ Central angle: ∠LOP

✔ Answer: ∠LOP

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[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

[7]

Angle: ∠POR
→ Rays OP and OR → intercepts arc PR → Arc PR

✔ Answer: Arc PR

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[8]

Angle: ∠AOB
→ Rays OA and OB → intercepts arc AB → Arc AB

✔ Answer: Arc AB

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[9]

Angle: ∠WOX
→ Rays OW and OX → intercepts arc WX → Arc WX

✔ Answer: Arc WX

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## Part 4: Find the measure of the arc indicated

[10]

Given: Central angle ∠AOB = 120°
→ Measure of arc AB = measure of central angle → 120°

✔ Answer: 120°

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[11]

Given: Central angle ∠AOB = 144°
→ Measure of arc AB = 144°
*(The radius 5 ft is irrelevant for arc measure — it’s about degrees, not length.)*

✔ Answer: 144°

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[12]

Given: Central angle ∠XOY = 65°
→ Measure of arc XY = 65°
*(Again, radius 4 ft is extra info — not needed for degree measure.)*

✔ Answer: 65°

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## ✔ Final Answers:

1. Arc HG
2. Arc HI
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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## 💡 Bonus Tip:
If you’re asked for arc length (not just measure), then you’d use the formula:
> Arc Length = (θ/360) × 2πr
where θ is the central angle in degrees, and r is the radius.

But here, all problems ask for measure of the arc, which is just the degree measure of the central angle.

Let me know if you want to calculate arc lengths too!

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✔ All done!