Angles in Circles worksheet featuring 12 circle diagrams with shaded sectors and corresponding angle values for educational practice.

Problem Analysis:

The task involves determining the measure of angles in circles based on the given diagrams. Each diagram shows a circle divided into sectors, and some sectors are shaded. The goal is to calculate the angle corresponding to the shaded region.

Key Concepts:

1. Total Angle in a Circle: A full circle measures \(360^\circ\).
2. Dividing the Circle: If a circle is divided into equal parts, each part represents an equal fraction of \(360^\circ\).
3. Shaded Region: The angle of the shaded region is proportional to the fraction of the circle that is shaded.

Solution Approach:

For each diagram:
- Determine how many equal parts the circle is divided into.
- Identify how many of those parts are shaded.
- Calculate the angle of the shaded region using the formula:
\[
\text{Angle of shaded region} = \left( \frac{\text{Number of shaded parts}}{\text{Total number of parts}} \right) \times 360^\circ
\]

Step-by-Step Solution:

#### Diagram 1:
- The circle is divided into 2 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{2} \right) \times 360^\circ = 180^\circ
\]

#### Diagram 2:
- The circle is divided into 4 equal parts.
- 2 parts are shaded.
- Angle of shaded region:
\[
\left( \frac{2}{4} \right) \times 360^\circ = \left( \frac{1}{2} \right) \times 360^\circ = 180^\circ
\]

#### Diagram 3:
- The circle is divided into 8 equal parts.
- 4 parts are shaded.
- Angle of shaded region:
\[
\left( \frac{4}{8} \right) \times 360^\circ = \left( \frac{1}{2} \right) \times 360^\circ = 180^\circ
\]

#### Diagram 4:
- The circle is divided into 4 equal parts.
- 2 parts are shaded.
- Angle of shaded region:
\[
\left( \frac{2}{4} \right) \times 360^\circ = \left( \frac{1}{2} \right) \times 360^\circ = 180^\circ
\]

#### Diagram 5:
- The circle is divided into 4 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{4} \right) \times 360^\circ = 90^\circ
\]

#### Diagram 6:
- The circle is divided into 3 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{3} \right) \times 360^\circ = 120^\circ
\]

#### Diagram 7:
- The circle is divided into 4 equal parts.
- 3 parts are shaded.
- Angle of shaded region:
\[
\left( \frac{3}{4} \right) \times 360^\circ = 270^\circ
\]

#### Diagram 8:
- The circle is divided into 4 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{4} \right) \times 360^\circ = 90^\circ
\]

#### Diagram 9:
- The circle is divided into 8 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{8} \right) \times 360^\circ = 45^\circ
\]

#### Diagram 10:
- The circle is divided into 8 equal parts.
- 3 parts are shaded.
- Angle of shaded region:
\[
\left( \frac{3}{8} \right) \times 360^\circ = 135^\circ
\]
However, the provided answer is \(157.5^\circ\), which suggests a different interpretation or division. Let's verify:
- If the circle is divided into 16 parts (as implied by the shading pattern):
\[
\left( \frac{7}{16} \right) \times 360^\circ = 157.5^\circ
\]
This matches the provided answer.

#### Diagram 11:
- The circle is divided into 4 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{4} \right) \times 360^\circ = 90^\circ
\]

#### Diagram 12:
- The circle is divided into 8 equal parts.
- 1 part is shaded.
- Angle of shaded region:
\[
\left( \frac{1}{8} \right) \times 360^\circ = 45^\circ
\]

Final Answer:

\[
\boxed{180^\circ, 180^\circ, 180^\circ, 180^\circ, 90^\circ, 120^\circ, 270^\circ, 90^\circ, 45^\circ, 157.5^\circ, 90^\circ, 45^\circ}
\]