The problem is asking to find $m \angle FHG$ in the diagram, given that:

• $m \angle CIH = 107^\circ$
• $m \angle BGD = 39^\circ$

Let's analyze this step by step:

1. $\triangle CIH$ is part of the configuration, where $m \angle CIH = 107^\circ$.
2. $\triangle BGD$ contains $m \angle BGD = 39^\circ$.
3. You are asked to find $m \angle FHG$.

Step-by-step Explanation:

We will assume the following properties based on the appearance of the diagram:

• Line $CG$ and line $AB$ seem to intersect at point $H$, suggesting that some angles are vertically opposite.
• $\angle FHG$ appears to be related to the other angles by either linear pair or angle sum properties in triangles or straight lines.

To calculate $m \angle FHG$, we likely need to utilize relationships between the angles formed by intersecting lines and supplementary angles. Without clear intersection properties labeled, it may also involve recognizing corresponding or vertically opposite angles.

However, I would need to confirm more about how the points, angles, and lines are interconnected geometrically. Would you like a more detailed geometric breakdown based on these assumptions, or should I clarify specific parts of the diagram for you?

Here are five follow-up questions to explore this further:

1. Do the lines appear to be parallel or intersecting, and how might that impact the angle relationships?
2. What geometric theorems could apply to this configuration (e.g., vertically opposite angles, angle sum in triangles)?
3. Are there any additional given angle measurements in the full problem that are not in the current image?
4. How can supplementary angles help solve for $m \angle FHG$?
5. Could the triangle angle sum property play a role here?

Tip: When dealing with intersecting lines, always check for vertically opposite and supplementary angles to help simplify calculations.