In a triangle, the incenter is the point where the angle bisectors of the angles intersect, and it is also the center of the incircle. The important property of tangents to a circle from a point outside the circle is that they are equal in length.
Given that \( D \) is the incenter of triangle \( BCA \), let’s analyze the situation with the given \( m\angle FDG = 136^\circ \).
Since \( D \) is the incenter, angles \( FDG \) and \( HDF \) sum up to \( 180^\circ \) because they are on a straight line (line DG). Thus:
\[ m\angle FDG + m\angle HDF = 180^\circ \]
Substituting the value of \( m\angle FDG = 136^\circ \):
\[ 136^\circ + m\angle HDF = 180^\circ \]
To find \( m\angle HDF \):
\[ m\angle HDF = 180^\circ - 136^\circ = 44^\circ \]
Now, the measure of angle \( FHG \) is equal to angle \( HDF \) since segments \( FH \) and \( GD \) are both tangents to circle \( D \) from points \( F \) and \( G \). Thus:
\[ m\angle FHG = m\angle HDF = 44^\circ \]
So the final answer is:
\[ \boxed{44^\circ} \]
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