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Let's solve part (b) of Problem 20 from the provided image.

### Problem 20 (b)
We need to find the values of the angles around point O. The given angles are as follows:
- \( \angle AOD = 65^\circ \)
- \( \angle BOC = 90^\circ \)
- \( \angle EOF = 75^\circ \)

We know that the sum of all angles around a point is \( 360^\circ \).

First, let's identify the unknown angles. Let's denote them as:
- \( \angle COD = x \)
- \( \angle BOE = y \)
- \( \angle EOD = z \)

We can use the given information and the fact that adjacent angles around a point sum to \( 360^\circ \):

\[ \angle AOD + \angle COD + \angle BOE + \angle EOF + \angle FOE = 360^\circ \]

Plugging in the known values:

\[ 65^\circ + x + 90^\circ + 75^\circ + y = 360^\circ \]

Since \( \angle BOE \) and \( \angle EOF \) form a straight line, they sum up to \( 180^\circ \):

\[ 90^\circ + y = 180^\circ \]
\[ y = 180^\circ - 90^\circ \]
\[ y = 90^\circ \]

Now let's sum all the known angles:

\[ 65^\circ + 90^\circ + 75^\circ + 90^\circ + z = 360^\circ \]

Summing these:

\[ 65 + 90 + 75 + 90 = 320 \]
\[ 320 + z = 360 \]
\[ z = 360 - 320 \]
\[ z = 40^\circ \]

Therefore:
- \( \angle COD = 90^\circ \)
- \( \angle BOE = 90^\circ \)
- \( \angle EOF = 75^\circ \)
- \( \angle EOD = 40^\circ \)

So, the angles around point O are correctly identified and \( \angle EOD \) is \( 40^\circ \).

Would you like more details on any of these solutions, or do you have any questions?

### Follow-up Questions
1. How would the solution change if \( \angle AOD \) was \( 75^\circ \) instead of \( 65^\circ \)?
2. What if \( \angle BOC \) was \( 120^\circ \)? How would you solve for the remaining angles?
3. Can you find the value of \( x \) if \( \angle EOF = 85^\circ \)?
4. What is the general formula for the sum of angles around a point?
5. How can you verify the correctness of your calculated angles in this problem?
6. What would be the effect of changing one of the angles to \( 100^\circ \)?
7. Can you derive a formula to find an unknown angle around a point given all other angles?
8. What if you know that one of the angles is supplementary to another? How does this affect the calculation?

### Tip
Always remember that the sum of angles around a point is \( 360^\circ \). This fundamental property is crucial in solving problems involving angles around a point.

Explore the solution to finding angles around point O in geometry. Learn about angle properties and calculations involved in solving for specific angle values around a point. Suitable for high school students and geometry enthusiasts.

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