To solve for $z$ in this geometric problem, let's analyze the given diagram step by step:

Step-by-Step Solution:

1. Key Observations:

   • The figure is a circle with a set of chords forming an inscribed angle $z$.
   • There is a given angle of $40^\circ$, which appears to be an inscribed angle as well.
   • Inscribed angles that subtend the same arc are equal.

2. Inscribed Angle Theorem:

   • According to the theorem, an inscribed angle is half the measure of the arc it subtends.
   • Both the given $40^\circ$ angle and $z$ subtend the same arc (the major arc at the bottom of the circle).

3. Conclusion: Since the angles subtend the same arc, they must be equal. Therefore: $z = 40^\circ$

Final Answer:

$z = 40^\circ$

Would you like a detailed explanation of the inscribed angle theorem or any related concepts? Let me know!

Related Questions:

1. What is the relationship between inscribed angles subtending the same arc in a circle?
2. How do you calculate the arc measure from an inscribed angle in a circle?
3. Can an inscribed angle subtend a semicircle? What would its measure be?
4. How do central angles relate to inscribed angles in a circle?
5. How can you use the inscribed angle theorem to prove other circle properties?

Tip:

In problems involving circles, always check if the angles subtend the same arc. This can simplify the problem significantly!