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To solve for the values of \( a \), \( b \), \( c \), and \( d \) in this cyclic quadrilateral, we will use several geometric theorems, particularly the **angle in the same segment theorem** and the **opposite angles of a cyclic quadrilateral sum to 180°**.

### Step-by-step Justification:
1. **Angle \( a^\circ \) (at \( W \)):**
   Using the **angles in the same segment** theorem, we know that angles subtended by the same arc are equal. Here, angle \( a^\circ \) subtended by arc \( XZ \) is equal to angle \( 60^\circ \) at \( X \) (since both are subtended by the arc \( ZW \)). Therefore,
   \[
   a^\circ = 60^\circ
   \]

2. **Angle \( b^\circ \) (at \( W \)):**
   Again using the **angles in the same segment** theorem, we know that angle \( b^\circ \) subtended by arc \( XY \) is equal to angle \( 24^\circ \) at \( X \) (since both are subtended by the arc \( YW \)). Therefore,
   \[
   b^\circ = 24^\circ
   \]

3. **Angle \( c^\circ \) (at \( X \)):**
   Since we have a cyclic quadrilateral, the opposite angles must sum to \( 180^\circ \). Angle \( c^\circ \) is opposite to the given angle \( 39^\circ \) at \( Z \). Therefore, we can calculate \( c^\circ \) as:
   \[
   c^\circ = 180^\circ - 39^\circ = 141^\circ
   \]

4. **Angle \( d^\circ \) (at \( Y \)):**
   Using the same cyclic quadrilateral property, angle \( d^\circ \) is opposite to the angle \( 57^\circ \) at \( Z \), and so we calculate \( d^\circ \) as:
   \[
   d^\circ = 180^\circ - 57^\circ = 123^\circ
   \]

### Final Values:
- \( a = 60^\circ \)
- \( b = 24^\circ \)
- \( c = 141^\circ \)
- \( d = 123^\circ \)

Let me know if you'd like more details on any step!

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Here are 5 related questions you might find interesting:
1. How do you prove that angles in the same segment are equal?
2. Can the sum of the opposite angles in any quadrilateral exceed 360°?
3. What are some real-world applications of cyclic quadrilaterals?
4. How can you construct a cyclic quadrilateral using a compass and straightedge?
5. How do the properties of cyclic quadrilaterals change in 3D geometry?

**Tip:** Always remember that angles subtended by the same arc in a circle are equal, a key fact used frequently in circle theorems!

Work out the values of a, b, c, and d in a cyclic quadrilateral. Justify each of your answers.

Learn how to solve for angles in a cyclic quadrilateral using key theorems. This step-by-step guide covers circle geometry concepts like angles in the same segment and the cyclic quadrilateral theorem. Suitable for students in Grades 9-11.

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