This problem involves a triangle $ABC$ inscribed in a circle, with $O$ being the center of the circle. The given angle is $\angle BCA = 82^\circ$, and we are tasked with finding the central angle $\angle BOA$.

Step-by-Step Solution:

1. Key property of inscribed and central angles:
   For any arc of a circle, the central angle is twice the inscribed angle subtending the same arc.
   In this case, $\angle BOA$ subtends the same arc as $\angle BCA$.

2. Relation between $\angle BOA$ and $\angle BCA$:
   $\angle BOA = 2 \cdot \angle BCA$

3. Substitute the given value of $\angle BCA = 82^\circ$:
   $\angle BOA = 2 \cdot 82^\circ = 164^\circ$

Final Answer:

$\boxed{\angle BOA = 164^\circ}$

Would you like a deeper explanation of this property or other examples?

Follow-up Questions:

1. What is the relation between central and inscribed angles in a circle?
2. How does the concept of subtended arcs apply to angles in circles?
3. What happens if $\angle BCA$ were obtuse or acute?
4. Could you calculate $\angle BOA$ if the angle at the center were given first?
5. What other problems involve triangles inscribed in circles?

Tip:

Remember that central angles are always twice the inscribed angles subtending the same arc! This rule is fundamental for solving problems involving circles and angles.