To solve for the measure of angle $\angle BAC$ in this circle geometry problem, let's analyze the information given.

1. Understanding the Triangle in a Circle:

   • The triangle $\triangle BOA$ is inscribed in the circle with the center $O$.
   • $\angle BOA$, which is at the center, is given as $85^\circ$.

2. Properties of Central and Inscribed Angles:

   • In a circle, an inscribed angle is half the measure of the central angle that subtends the same arc.
   • Here, $\angle BAC$ is an inscribed angle that subtends the same arc $BC$ as the central angle $\angle BOA$.

3. Calculating $\angle BAC$:

   • Since $\angle BAC$ is an inscribed angle that subtends the same arc as $\angle BOA$, we can use the formula: $m\angle BAC = \frac{1}{2} m\angle BOA$
   • Substituting $85^\circ$ for $m\angle BOA$: $m\angle BAC = \frac{1}{2} \times 85^\circ = 42.5^\circ$

4. Answer:

   • The measure of $\angle BAC$ is $42.5^\circ$, which corresponds to option c.

Would you like a more detailed explanation of any step, or have additional questions?

Here are some related questions for further exploration:

1. How would the solution change if $\angle BOA$ were a different value?
2. What if $BAC$ were not an inscribed angle? How would that affect the calculation?
3. Can you explain why inscribed angles are always half of the central angle subtending the same arc?
4. What is the relationship between other inscribed angles subtending the same arc?
5. How does this principle apply to cyclic quadrilaterals?

Tip: In circle geometry, always remember that inscribed angles subtending the same arc are equal.