Let's solve this problem by focusing on the angles given in the figure.

1. We are given that $AOB$ is a straight line, so the angles on the line sum to $180^\circ$.
2. Angle $\angle AOC = x^\circ$, and the other angles around point $O$ include:
   • $65^\circ$ for $\angle COD$,
   • $(2x - 20)^\circ$ for $\angle BOD$.

Step-by-step process:

1. The angles $\angle AOC$, $\angle COD$, and $\angle BOD$ are all adjacent, forming a straight line at $AOB$. Thus, we have the equation:

$x + 65^\circ + (2x - 20^\circ) = 180^\circ$

2. Simplifying this equation: $x + 65 + 2x - 20 = 180$ $3x + 45 = 180$ $3x = 135$ $x = 45^\circ$

Now, we can find the required angles:

• $\angle AOC = x = 45^\circ$
• $\angle BOD = 2x - 20 = 2(45^\circ) - 20^\circ = 90^\circ - 20^\circ = 70^\circ$

Final answers:

• $\angle AOC = 45^\circ$
• $\angle BOD = 70^\circ$

Would you like a deeper explanation of any part of this solution?

Here are five related questions you might find useful:

1. What is the relationship between angles on a straight line?
2. How do adjacent angles behave around a point?
3. How can we apply algebraic techniques to solve angle problems like this?
4. Can you think of other geometric situations where angles sum to 180 degrees?
5. How would the problem change if $AOB$ wasn't a straight line?

Tip: Always check if angles form a linear pair or a full circle to simplify the calculation.