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!

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!