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To solve this question, let's analyze the information provided and apply circle geometry theorems.

1. **Information Provided**:
   - **Given angles**:
     - \( \angle ABF = 25^\circ \)
     - \( \angle EAD = 33^\circ \)
   - **Chords**: \( EF = ED \)
   - Points \( A, B, C, D, F \) lie on the circle with center \( O \).
   - We need to determine the sizes of angles \( \hat{B_1} \), \( \hat{C_2} \), \( \hat{E_2} \), and \( \hat{F_1} \) with reasoning.

2. **Solution**:

   Let's address each part of the question.

   ### 9.1: \( \hat{B_1} \)
   - Since \( \angle EAD = 33^\circ \) is given, and by the **Inscribed Angle Theorem**, the angle subtended by the same arc \( ED \) on the opposite side of the circle will also be equal to \( 33^\circ \).
   - Thus, \( \hat{B_1} = 33^\circ \).

   ### 9.2: \( \hat{C_2} \)
   - By the **Alternate Segment Theorem**, the angle between the chord and the tangent at one end of the chord is equal to the angle in the alternate segment.
   - Since \( \angle ABF = 25^\circ \), and \( C_2 \) is in the alternate segment, \( \hat{C_2} = 25^\circ \).

   ### 9.3: \( \hat{E_2} \)
   - Given that chords \( EF = ED \), we have an **isosceles triangle \( \triangle EFD \)**.
   - Therefore, the base angles are equal. Since \( \hat{B_1} = 33^\circ \), it means that \( \hat{E_2} = 33^\circ \).

   ### 9.4: \( \hat{F_1} \)
   - By the **Inscribed Angle Theorem**, angles subtended by the same arc (opposite sides of the circle) are equal.
   - Therefore, \( \hat{F_1} = \angle ABF = 25^\circ \).

These are the angle measures with reasons for each.

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Would you like a deeper explanation of any specific part? Here are some related questions you might find helpful:

1. What is the Inscribed Angle Theorem, and how does it apply to circles?
2. How does the Alternate Segment Theorem work with chords and tangents?
3. What properties do isosceles triangles within circles have?
4. How can we identify angles subtended by the same arc?
5. What are other ways to find angles in circle geometry?

**Tip**: In circle geometry, identifying arcs and corresponding inscribed angles is often key to solving angle-related problems.

In the diagram below, A, B, C, D, and F are points on the circle. O is the center of the circle. Chords EF = ED, ∠ABF = 25° and ∠EAD = 33°. Determine, with reasons, the size of the following: ∠B₁, ∠C₂, ∠E₂, and ∠F₁.

Learn how to determine angles in circle geometry using inscribed angle and alternate segment theorems. This problem involves calculating angles ∠B₁, ∠C₂, ∠E₂, and ∠F₁ with reasoning, suitable for high school students studying advanced geometry.

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