The problem asks to identify a central angle in a circle with center $F$. $GI$ is a diameter and the measure of angle $GFH$ is $100^\circ$.

1. Problem Description

The problem asks to identify a central angle in a circle with center $F$. $GI$ is a diameter and the measure of angle $GFH$ is $100^\circ$.

2. Solution Steps

A central angle is an angle whose vertex is the center of the circle. In the given circle, the center is point $F$. We are given that $\angle GFH = 100^\circ$.

Another possible central angle is $\angle GFI$.

Since $GI$ is a diameter, $\angle GFI$ is a straight angle, and its measure is $180^\circ$.

Also, since $GI$ is a diameter, we know that $\angle IFH$ is supplementary to $\angle GFH$. Thus,

$m\angle IFH = 180^\circ - m\angle GFH = 180^\circ - 100^\circ = 80^\circ$.

$\angle IFH$ is a central angle.

3. Final Answer

$\angle GFH$ or $\angle IFH$ or $\angle GFI$

I will choose $\angle GFH$.