In this problem, we need to find the value of $b$ in the given geometric figure where $POQ$ is a straight line.

Given Information:

• $POQ$ is a straight line, meaning the angle formed by $POQ$ is $180^\circ$.
• There are three angles at point $O$: $b^\circ$, $b^\circ$, and $4b^\circ$.

Solution Approach:

Since $POQ$ is a straight line, the sum of all the angles along this line must be equal to $180^\circ$. Therefore, we can write the equation as: $b + b + 4b = 180^\circ$

Simplify the equation: $6b = 180^\circ$

Solve for $b$: $b = \frac{180^\circ}{6} = 30^\circ$

Conclusion:

The value of $b$ is $30^\circ$.

Would you like more details or have any questions on the steps? Here are some related questions you might consider:

1. What other methods can be used to verify the value of $b$ in this scenario?
2. How do we know that the angles must add up to $180^\circ$?
3. What are some common properties of straight angles in geometric figures?
4. How would the problem change if the angles had different values?
5. How can we apply the concept of supplementary angles to solve similar problems?

Tip: In problems involving straight lines, always remember that the angles on a straight line add up to $180^\circ$.