The problem given in the image involves a circle with center $O$. The task is to calculate the size of the angle $\angle BOC$, given that $\angle ADB = 70^\circ$ and $\angle AOC = 74^\circ$.

Here’s the breakdown:

1. Explanation of key parts:

   • $O$ is the center of the circle.
   • $\angle ADB = 70^\circ$ is an inscribed angle.
   • $\angle AOC = 74^\circ$ is a central angle subtended by the arc $AC$.
   • We are asked to find the value of $\angle BOC$.

2. Understanding the Relationship:

   • The inscribed angle $\angle ADB$ subtends the same arc as the central angle $\angle AOB$.
   • The inscribed angle is always half the central angle subtending the same arc. This means: $\angle AOB = 2 \times \angle ADB = 2 \times 70^\circ = 140^\circ$

3. Finding $\angle BOC$:

   • The total angle around the center $O$ on a circle is $360^\circ$. So: $\angle BOC = 360^\circ - (\angle AOB + \angle AOC)$ Substituting the values: $\angle BOC = 360^\circ - (140^\circ + 74^\circ) = 360^\circ - 214^\circ = 146^\circ$

Thus, the measure of $\angle BOC$ is 146°.

Would you like more details on this solution, or have any other questions?

Here are five related questions:

1. How is the relationship between central and inscribed angles derived?
2. What other methods can be used to find angles in a circle geometry problem?
3. How can you prove that the sum of angles around a point is $360^\circ$?
4. If $\angle AOC$ were increased, how would $\angle BOC$ change?
5. What are some real-world applications of circle geometry?

Tip: Remember that inscribed angles are always half of the central angle subtending the same arc in a circle.