Determine with reasons the size of: 3.1 3.2 3.3 3.4 3.5 3.6 𝐷 2 𝐵 2 𝐸 2 𝐸 1 𝐶𝑂 𝐸𝐶 𝐵 𝐵

To determine the sizes of the angles or segments labeled in the problem, let's analyze each part based on the diagram you provided. I'll address each point mentioned (e.g., \( D2 \), \( B2 \), etc.) and give the reasoning:

### 3.1 \( D2 \) (likely referring to \( \angle DAB \))
- **Given:** \( \angle DAB = 22^\circ \).
- **Reason:** This angle is directly provided in the diagram.

### 3.2 \( B2 \)
- **Interpretation:** The angle \( \angle BAC \) or any other angle involving \( B \) near the tangent.
- **Reason:** If this is \( \angle BAC \), and assuming that \( AB \) is tangent to the circle at \( A \), \( \angle BAC \) could be determined using angle relationships in circles, specifically using the alternate segment theorem.

### 3.3 \( E2 \)
- **Interpretation:** If \( E2 \) refers to an angle at \( E \) or involving \( E \), we might need more context.
- **Reason:** The angle at \( E \) could involve \( \angle DEB \) or any other relevant angle.

### 3.4 \( E1 \)
- **Interpretation:** This could be referring to \( \angle EAC \) or any other relevant angle involving \( E \).
- **Reason:** Depending on the circle's properties and given angles, this angle can be calculated using geometric properties of circles.

### 3.5 \( CO \) (likely referring to the angle \( \angle COE \))
- **Interpretation:** The angle subtended at the center \( O \) of the circle.
- **Reason:** If \( \angle COE \) is needed, it could be calculated based on the known arc or using relationships between angles at the center and the circumference.

### 3.6 \( EC \)
- **Interpretation:** This could refer to the length of segment \( EC \) or the angle \( \angle EAC \).
- **Reason:** Lengths could be calculated using trigonometry or circle theorems, while angles could be deduced from known properties of the circle.

### General Strategy:
1. **Tangent-Secant Theorem:** For a tangent and a secant from a common external point \( B \), the tangent squared equals the product of the external segment of the secant and the entire secant.
2. **Angle Properties in Circles:**
   - The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
   - Angles subtended by the same arc at the circumference are equal.

Would you like to explore a specific part or need further clarification on any of these elements? 

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**Related Questions:**

1. How would you find \( \angle C \) in this context?
2. What is the sum of opposite angles in a cyclic quadrilateral?
3. Can you derive the length of a tangent from a point outside the circle?
4. How does the angle at the center compare with the angle at the circumference subtended by the same arc?
5. What theorem can help you determine the angle between a tangent and a chord?

**Tip:** Remember, the angle at the center of a circle is twice the angle subtended at the circumference by the same arc.

Explore how to determine the sizes of angles and segments labeled in a circle geometry problem. This detailed analysis covers various theorems and formulas, including the Tangent-Secant Theorem, Alternate Segment Theorem, and angle properties in circles. Suitable for high school students studying geometry.

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