Mrs. Newells Math: #MTBoS30: Central Angles and Arcs - Free Printable

Let’s solve the problem step by step using the given circle diagram with center E, diameter NL, and point W on the circle such that ∠NEW = 22°.

---

Key Concepts:

- A minor arc is an arc less than 180°.
- A major arc is an arc greater than 180°.
- A semicircle is exactly 180° (half the circle).
- The measure of an arc equals the measure of its central angle.
- Since NL is a diameter, it divides the circle into two semicircles: arc NKL and arc NWL (each 180°).

---

Given:

- ∠NEW = 22° → so arc NW = 22° (central angle equals arc measure).
- KE ⊥ NL → so ∠NEL = 90° → arc KL = 90°.
- NL is diameter → arc NKL = 180°, arc NWL = 180°.

---

Now let’s find each arc:

---

① mNW

- Central angle: ∠NEW = 22° → arc NW = 22°
- Since 22° < 180° → Minor arc

✔ Answer: Minor arc, 22°

---

② mKNL

- This is the arc from K to N to L — going through N.
- Since NL is diameter, and K is at the top, this arc goes from K → N → L.
- Arc KN = ? Let's compute:
- From K to N: since KE ⊥ NL, arc KL = 90°, so arc KN = 180° - 90° = 90°? Wait — no.

Actually, let’s think carefully:

Points in order around circle: starting from N, going clockwise: N → W → L → K → N? Or counterclockwise?

Looking at diagram: K is top, N is left, L is right, W is between N and L on the bottom arc.

So order (clockwise): N → W → L → K → N

But arc KNL means from K to N to L — that would be going K → N → L.

From K to N: that’s going counterclockwise from K to N — which passes through the top-left? But since KE ⊥ NL, and NL is horizontal, K is directly above E.

So arc from K to N: since ∠KEN = 90° (because KE ⊥ NL), arc KN = 90°.

Then from N to L is the diameter — 180°? No — arc NL is 180°, but we’re going from K to N to L — so total arc KNL = arc KN + arc NL?

Wait — that doesn’t make sense because N to L is already 180°, and adding KN would go over 180°.

Actually, arc KNL is the arc from K to L passing through N. So it’s the long way around — major arc.

Total circle = 360°. The minor arc from K to L is 90° (since ∠KEL = 90°). So major arc KNL = 360° - 90° = 270°

✔ Answer: Major arc, 270°

---

③ mKL

- Central angle ∠KEL = 90° (since KE ⊥ NL) → arc KL = 90°
- 90° < 180° → Minor arc

✔ Answer: Minor arc, 90°

---

④ mNKL

- This is arc from N to K to L — meaning going from N → K → L.
- Since NL is diameter, and K is at top, this arc goes from N up to K then down to L — that’s half the circle? Actually, yes!

Arc NKL: from N to L passing through K. Since NL is diameter, and K is on the circle perpendicular to it, this arc is exactly the top semicircle.

Measure: 180° → Semicircle

✔ Answer: Semicircle, 180°

---

⑤ mKW

- Need to find arc from K to W.
- We know:
- arc KL = 90°
- arc LW = ? From L to W: since arc NW = 22°, and N to L is 180°, then arc WL = 180° - 22° = 158°? Wait — no.

Actually, points: N → W → L → K → N (clockwise)

So from K to W clockwise: K → N → W → that’s arc KN + arc NW.

We know arc KN = 90° (since ∠KEN = 90°), arc NW = 22° → total = 90° + 22° = 112°

Alternatively, from K to W counterclockwise: K → L → W → arc KL + arc LW.

arc KL = 90°, arc LW = ?

Since arc NW = 22°, and arc NL = 180°, then arc WL = 180° - 22° = 158°? No — arc from W to L is part of the lower semicircle.

Actually, arc from N to W is 22°, so from W to L is 180° - 22° = 158°

So arc K to W counterclockwise: K → L → W = 90° + 158° = 248° — that’s the major arc.

But typically, unless specified, we take the minor arc when just saying "arc KW".

So minor arc KW is the smaller one: 112° (K→N→W) vs 248° (K→L→W) → so minor is 112°.

Is 112° < 180°? Yes → Minor arc

✔ Answer: Minor arc, 112°

---

⑥ mKLW

- Arc from K to L to W — meaning K → L → W.
- That’s arc KL + arc LW.
- arc KL = 90°
- arc LW = 180° - 22° = 158° (since arc NW = 22°, and N to L is 180°)
- Total = 90° + 158° = 248°
- > 180° → Major arc

✔ Answer: Major arc, 248°

---

⑦ mLKW

- Arc from L to K to W — L → K → W.
- From L to K: arc LK = 90° (same as KL, just reverse direction)
- From K to W: we calculated minor arc KW = 112°, but that’s K→N→W. Here we’re going K→W via N? Actually, from K to W via N is 112°, but if we go L → K → W, that’s L → K → N → W? That would be arc LK + arc KN + arc NW = 90° + 90° + 22° = 202°? That seems too big.

Wait — better to think: total circle = 360°.

Arc LKW: from L to W passing through K.

The other way from L to W is directly along the bottom: arc LW = 158° (as before).

So arc LKW = 360° - 158° = 202°

> 180° → Major arc

✔ Answer: Major arc, 202°

---

⑧ mNKW

- Arc from N to K to W — N → K → W.
- From N to K: arc NK = 90° (since ∠NEK = 90°)
- From K to W: minor arc KW = 112°? But that’s K→N→W — which would double back.

Actually, from N to K to W: N → K → W.

From N to K is 90° (counterclockwise).

From K to W: if we go K → W without going back to N, that’s the long way? No — in the circle, from K to W can go clockwise or counterclockwise.

But arc NKW implies going from N to K to W — so likely N → K → W clockwise? That would be N → K → L → W? That’s arc NK + arc KL + arc LW = 90° + 90° + 158° = 338° — too big.

Better: the arc N to W not passing through L is 22°, but here we’re going through K.

So arc NKW = arc NK + arc KW, where arc KW is the part from K to W not passing through N.

But since K to W has two paths: short (112° via N) and long (248° via L).

If we go N → K → W, and we don’t want to backtrack, we must go N → K → L → W? That’s 90° + 90° + 158° = 338° — major arc.

Alternatively, perhaps it’s intended to be the arc from N to W passing through K — which is the long way.

Since minor arc NW is 22°, the major arc NKW = 360° - 22° = 338°

Yes — that makes sense.

✔ Answer: Major arc, 338°

---

## ✔ Final Answers:

| # | Arc | Type | Measure |
|---|----------|------------|---------|
| 1 | mNW | Minor arc | 22° |
| 2 | mKNL | Major arc | 270° |
| 3 | mKL | Minor arc | 90° |
| 4 | mNKL | Semicircle | 180° |
| 5 | mKW | Minor arc | 112° |
| 6 | mKLW | Major arc | 248° |
| 7 | mLKW | Major arc | 202° |
| 8 | mNKW | Major arc | 338° |

---

## 📌 Summary:
- Use central angles to find arc measures.
- Remember: minor < 180°, major > 180°, semicircle = 180°.
- Diameter NL creates two 180° arcs.
- Add/subtract known arcs to find unknowns.

Let me know if you’d like a diagram sketch or further clarification!