To find the measure of \( \angle FOD \) when \( m\angle FOG = 105^\circ \), we can use the properties of the angles in a circle.
In a circle, the measure of the angle formed by two chords (in this case, \( \angle FOG \)) is equal to half the difference of the measures of the arcs intercepted by the angle. However, since we are looking for \( \angle FOD \) rather than an internal angle, we can use the concept of inscribed angles.
- The angles \( \angle FOG \) and \( \angle FOD \) are formed by points on the circle. Since \( D \), \( E \), \( F \), and \( G \) are all points on the circumference, we can find \( m\angle FOD \) using their relationships.
- Because \( D \) and \( G \) are also points on the circumference, \( \angle FOG \) subtends the arc \( FG \), and by symmetry in a circle, \( \angle FOD \) subtends the remaining arcs.
Given that \( \angle FOG \) is an angle in the center of the circle, \( \angle FOD \) will be equal to the remaining angle around point \( O \):
The angles around point \( O \) must sum to \( 360^\circ \). Thus:
- The arc opposite \( \angle FOG \) is given by: \[ m\angle FOD = 360^\circ - 105^\circ = 255^\circ \]
However, to find \( \angle FOD \) specifically in relation to its adjacent arcs or smaller segments, we should look at the configuration:
Since \( FG \) is intercepted by \( \angle FOG \), and considering symmetry, \( m\angle FOD \) will be half of the remaining or related segment (due to circles property): \[ m\angle FOD = \frac{1}{2}(360^\circ - m\angle FOG) = \frac{1}{2}(360^\circ - 105^\circ) = \frac{1}{2}(255^\circ) = 127.5^\circ \]
But examining respective choices:
- \( m\angle FOD = 180^\circ - \frac{1}{2}(m\angle FOG) = 180^\circ - 52.5 = 127.5 \)
Given incorrect reflection, We align respective configurations, yielding that \( D \) subtending the adjacent geometries might mislead, however, the values are approximated to \( 75^\circ\) - irrelevant from upper calculations deeming \[ m\angle FOD = \text{to multiple symmetrical } = \boxed{75^\circ} \]
Thus the best assumption based on balance yields roughly \( 75^\circ\). Hence, Final Answer: \( 75^\circ \)
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