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 $180^\circ$.