To find the measure of angle \( \angle FOD \) given that \( m\angle FOG = 105^\circ \), we can use the fact that the angle formed by two chords \( \angle FOG \) at the center of the circle is twice the angle that subtends the same arc at any point on the circumference of the circle.
In this case:
\[ m\angle FOD = \frac{1}{2} \cdot m\angle FOG \]
Thus:
\[ m\angle FOD = \frac{1}{2} \cdot 105^\circ = 52.5^\circ \]
Since none of the provided options match \( 52.5^\circ \), it appears there may have been a misunderstanding in determining the arcs involved or a typographical error regarding the potential answer choices.
However, if you mean the angle subtended by the same arc \( FG \) at a different point along the circle's circumference (like \( D \)), we often use the formula where the angle subtended at the center is twice that at the circumference.
If \( O \) is the center and you want \( \angle FOD \) directly (which subtends the same arc as \( FOG \)), the angles would indeed relate simply like stated. Therefore, if the options provided do not match our calculated angle of \( 52.5^\circ \), there's likely either confusion regarding the arcs or an error in the choices.
To summarize:
- \( m\angle FOD = 52.5^\circ \) from \( m\angle FOG = 105^\circ \).
- All provided choices (85, 75, 150, 37.5 degrees) are incorrect based on the angle calculations.
You should clarify with the initial problem or image for additional context or verify other angles affecting \( \angle FOD \).