Angles Worksheets | Arcs and Central Angles Worksheets - Free Printable

Problem Analysis and Solution

The worksheet focuses on understanding arcs and central angles in circles. Let's solve each part step by step.

---

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

1. Problem 1:
- Given: ∠HGK
- The central angle ∠HGK intercepts the arc HK.
- Answer: $\widehat{HK}$

2. Problem 2:
- Given: ∠QPR
- The central angle ∠QPR intercepts the arc QR.
- Answer: $\widehat{QR}$

---

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

3. Problem 3:
- Given: Arc $\widehat{MN}$
- The central angle that intercepts $\widehat{MN}$ is ∠MON.
- Answer: ∠MON

4. Problem 4:
- Given: Arc $\widehat{KM}$
- The central angle that intercepts $\widehat{KM}$ is ∠KHM.
- Answer: ∠KHM

---

#### Part 3: Find the measure of the arc or central angle indicated

5. Problem 5:
- Given: ∠VTU = 70°
- The measure of the arc $\widehat{VU}$ is equal to the measure of the central angle ∠VTU.
- Answer: $\widehat{VU} = 70^\circ$

6. Problem 6:
- Given: ∠EDG = 180°
- The measure of the arc $\widehat{EFG}$ is equal to the measure of the central angle ∠EDG.
- Answer: $\widehat{EFG} = 180^\circ$

---

#### Part 4: Solve for x

7. Problem 7:
- Given:
- $\widehat{KGH} = 275^\circ$
- ∠GFK = 155°
- ∠HFG = 2x
- Step 1: The measure of the entire circle is 360°. Since $\widehat{KGH} = 275^\circ$, the remaining arc $\widehat{KH}$ is:
\[
\widehat{KH} = 360^\circ - 275^\circ = 85^\circ
\]
- Step 2: The central angle ∠HFG intercepts the arc $\widehat{KH}$. Therefore, the measure of ∠HFG is equal to the measure of $\widehat{KH}$:
\[
\text{Measure of } \widehat{KH} = 85^\circ \implies \text{Measure of } \angle HFG = 85^\circ
\]
- Step 3: We are given that ∠HFG = 2x. So:
\[
2x = 85^\circ \implies x = \frac{85^\circ}{2} = 42.5^\circ
\]
- Answer: $x = 42.5^\circ$

8. Problem 8:
- Given:
- $\widehat{GF} = 100^\circ$
- ∠GEF = 4x
- Step 1: The central angle ∠GEF intercepts the arc $\widehat{GF}$. Therefore, the measure of ∠GEF is equal to the measure of $\widehat{GF}$:
\[
\text{Measure of } \widehat{GF} = 100^\circ \implies \text{Measure of } \angle GEF = 100^\circ
\]
- Step 2: We are given that ∠GEF = 4x. So:
\[
4x = 100^\circ \implies x = \frac{100^\circ}{4} = 25^\circ
\]
- Answer: $x = 25^\circ$

---

Final Answers:

1. $\boxed{\widehat{HK}}$
2. $\boxed{\widehat{QR}}$
3. $\boxed{\angle MON}$
4. $\boxed{\angle KHM}$
5. $\boxed{70^\circ}$
6. $\boxed{180^\circ}$
7. $\boxed{42.5^\circ}$
8. $\boxed{25^\circ}$

---

Summary:

- Arcs and Central Angles: The measure of an arc is equal to the measure of its corresponding central angle.
- Circle Properties: The total measure of a circle is 360°, which helps in finding missing arcs or angles.
- Solving for Variables: Use the relationships between arcs and central angles to set up equations and solve for unknowns.

$\boxed{\text{Final Answers Provided Above}}$