To solve this geometry problem involving the circle and its central angles, let's analyze the given information from the image:

Step 1: Understand the Problem

We are asked to find the measure of the angle $m \angle JFK$.

Step 2: Analyze the Given Data

• $m \angle GFI = 45^\circ$ (angle from points $G$ to $I$ passing through center $F$).
• $m \angle HFI = 83^\circ$ (angle from points $H$ to $I$ passing through center $F$).
• We are to assume that lines appearing to be diameters are indeed diameters.

Step 3: Use Properties of the Circle

• A diameter divides the circle into two semicircles, each measuring $180^\circ$.
• Since angles $\angle GFI$ and $\angle HFI$ are given as central angles, they represent the measures of their corresponding arcs on the circle.

Step 4: Calculate $m \angle JFK$

• The total measure of all angles around point $F$ is $360^\circ$.
• The central angles given sum up to: $45^\circ + 83^\circ = 128^\circ$
• The remaining angle around point $F$ can be found by subtracting the sum of the given angles from $360^\circ$: $360^\circ - 128^\circ = 232^\circ$
• Therefore, $m \angle JFK = 232^\circ$.

Conclusion

The measure of the angle $m \angle JFK$ is: $\boxed{232^\circ}$

Would you like further details on the solution, or do you have any specific questions?

Here are 5 related questions to expand your understanding:

1. How would you find the measure of an arc if only a portion of the circle is known?
2. What is the relationship between central angles and inscribed angles that subtend the same arc?
3. How does the presence of a diameter affect the measurement of arcs and central angles?
4. Can central angles be used to find the lengths of arcs if the circle's radius is known?
5. How would you calculate the area of a sector given its central angle?

Tip: Always remember that the sum of all central angles around a circle is $360^\circ$.