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In this problem, we are tasked with finding the value of \(\angle BOC\), given that points \(A\), \(O\), and \(B\) are collinear. Let’s break down the information:

- Points \(A\), \(O\), and \(B\) are collinear, meaning that line \(AB\) is a straight line.
- The given angles around point \(O\) are expressed as:
  - \(\angle AOD = x - 10^\circ\)
  - \(\angle DOC = 4x - 25^\circ\)
  - \(\angle BOC = x + 5^\circ\)

Since \(A\), \(O\), and \(B\) are collinear, the angle \(AOB\) is a straight angle, and the sum of angles on a straight line is \(180^\circ\). This gives us the equation:

\[
(x - 10^\circ) + (4x - 25^\circ) + (x + 5^\circ) = 180^\circ
\]

Now, solving for \(x\):

\[
(x - 10) + (4x - 25) + (x + 5) = 180
\]
\[
x + 4x + x - 10 - 25 + 5 = 180
\]
\[
6x - 30 = 180
\]
\[
6x = 210
\]
\[
x = 35^\circ
\]

Now that we have \(x = 35^\circ\), we can substitute this value into the expression for \(\angle BOC\):

\[
\angle BOC = x + 5^\circ = 35^\circ + 5^\circ = 40^\circ
\]

Thus, the value of \(\angle BOC\) is \(40^\circ\).

Would you like a more detailed breakdown of any part of the solution? Here are some related questions:

1. What is the reasoning behind summing the angles on a straight line?
2. Can you explain why \(A\), \(O\), and \(B\) being collinear makes \(AOB\) a straight angle?
3. How can I verify if the value of \(x\) is correct in this context?
4. What would happen if we used a different method to find \(x\)?
5. How do the angles \(\angle AOD\) and \(\angle DOC\) affect the overall calculation?

**Tip:** Always remember that the sum of angles around a point is \(360^\circ\), and the sum of angles on a straight line is \(180^\circ\).

In the given figure, find the value of ∠BOC, if points A, O, and B are collinear.

Solve for the value of ∠BOC in a geometric setup with collinear points A, O, and B. This problem involves algebraic expressions for angles, including (x - 10°), (4x - 25°), and (x + 5°), and applies the sum of angles on a straight line being 180°. Suitable for Grades 9-10.

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