In the diagram, we see a circle with a center point $C$, a central angle marked as $108^\circ$, and two inscribed angles labeled $m$ and $n$. The goal is likely to find the values of these angles $m$ and $n$.

Key geometric principles:

1. Inscribed Angle Theorem: An inscribed angle is half of the central angle subtended by the same arc.
2. Sum of angles around a point: The angles formed at a point (such as at the center of a circle) sum to $360^\circ$.

Steps:

1. The central angle given is $108^\circ$. This means that the arc subtended by this central angle is the same arc subtended by both inscribed angles $m$ and $n$.

2. By the Inscribed Angle Theorem, each inscribed angle $m$ and $n$ is half of the central angle that subtends the same arc. Thus, the measures of $m$ and $n$ are: $m = n = \frac{108^\circ}{2} = 54^\circ$

So, both angles $m$ and $n$ are $54^\circ$.

Would you like further explanation or have any questions?

Here are some related questions:

1. What is the relationship between central and inscribed angles in a circle?
2. How do you determine the measure of an inscribed angle from the central angle?
3. Can two different inscribed angles subtend the same arc in a circle?
4. What is the sum of all central angles around a point inside a circle?
5. How can the Inscribed Angle Theorem be applied to other shapes or arcs?

Tip: Always remember that the central angle is twice any inscribed angle subtended by the same arc.