I have viewed the image you uploaded. Let's solve the two parts of the problem:

Part (a):

We are given:

• $\angle XZY = 64^\circ$
• Arc $XY = 73^\circ$ We are asked to find the measure of arc $VW$.

Solution: The key fact here is that the measure of an inscribed angle is half of the arc it subtends. Here, $\angle XZY$ subtends arc $XY$, so: $\angle XZY = \frac{1}{2} \times \text{Arc } XY$ Substitute the values: $64^\circ = \frac{1}{2} \times 73^\circ$ This isn't correct, which implies that $\angle XZY$ subtends a different arc.

Instead, $\angle XZY$ subtends the arc opposite to arc $XY$, which is the rest of the circle. The total measure of the circle is $360^\circ$, so the remaining arc (arc $VW$) is: $\text{Arc } VW = 360^\circ - 73^\circ = 287^\circ$

Now, using the inscribed angle formula: $\angle XZY = \frac{1}{2} \times \text{Arc } VW$ Substitute the known value of $\angle XZY$: $64^\circ = \frac{1}{2} \times \text{Arc } VW$ Multiply both sides by 2: $128^\circ = \text{Arc } VW$

So, the measure of arc $VW = 128^\circ$.

Part (b):

We are given:

• $\angle ADB = 240^\circ$ We are asked to find $\angle ACB$.

Solution: In a circle, when two angles subtend the same arc, the angle at the circumference is half the angle at the center. Here, $\angle ADB$ is at the center, and it subtends arc $AB$. Therefore, the angle at the circumference $\angle ACB$ is: $\angle ACB = \frac{1}{2} \times \angle ADB$ Substitute the value of $\angle ADB$: $\angle ACB = \frac{1}{2} \times 240^\circ = 120^\circ$

Thus, $\angle ACB = 120^\circ$.

Let me know if you want further details or have any questions. Here are some related problems for you to explore:

1. What is the relationship between the angle at the center and the arc subtended by it?
2. How would the solution change if $\angle ADB$ was smaller than 180°?
3. What would happen if point $Z$ was moved closer to $X$ in part (a)?
4. Can you deduce a general rule for inscribed angles in any circle?
5. If you know arc $VW$, how would you calculate the corresponding inscribed angle?

Tip: Remember that angles at the circumference that subtend the same arc are always equal!