To solve this problem, let’s analyze the given information step by step.

Problem Analysis

1. Given Information:

   • $O$ is the center of the circle.
   • Line $AC$ is the diameter of the circle.
   • Angle $\angle BOA$ is $82^\circ$.

2. Goal:

   • Find the measure of $\angle BDC$.

Solution

Since $AC$ is the diameter, it divides the circle into two semicircles. Angle $\angle BDC$ subtends the diameter $AC$, meaning it lies on the circumference and spans half the circle.

According to the Inscribed Angle Theorem, an angle subtended by a diameter in a circle (i.e., an angle formed on the circumference by the diameter) is always a right angle. Thus: $\angle BDC = 90^\circ$

Conclusion

The measure of $\angle BDC$ is 90 degrees.

Would you like further explanation on this topic or have additional questions? Here are some related questions that might deepen your understanding:

1. Why does an inscribed angle subtended by a diameter always equal 90 degrees?
2. How would the solution change if $AC$ were not the diameter?
3. Can you apply the Inscribed Angle Theorem to other angles in the circle?
4. What is the relationship between central and inscribed angles in a circle?
5. How do we know that $\angle BOA$ is not directly related to $\angle BDC$?

Tip: Remember, an angle formed by a diameter and any point on the circle is always a right angle, due to the properties of the circle’s geometry.