In the given figure, we are asked to find the value of $x$, given that $AOB$ is a straight line.

Information from the figure:

1. $AOB$ is a straight line, meaning the sum of the angles on that line equals $180^\circ$.
2. The angle between $AOC$ is given as $(3x - 7)^\circ$.
3. The angle between $COD$ is given as $55^\circ$.
4. The angle between $DOB$ is given as $(x + 20)^\circ$.

Step-by-step solution:

Since $AOB$ is a straight line, the sum of the three angles must equal $180^\circ$. Therefore, we have the equation:

$(3x - 7) + 55 + (x + 20) = 180$

Now, let's simplify and solve for $x$:

$3x - 7 + 55 + x + 20 = 180$

Combine like terms:

$4x + 68 = 180$

Subtract 68 from both sides:

$4x = 112$

Divide by 4:

$x = 28$

Final Answer:

The value of $x$ is $28^\circ$.

Would you like more details or have any questions? Here are some related questions to further explore the concept:

1. What happens if one of the angles on the straight line changes?
2. How do we handle angles if the line was not straight but a different shape?
3. What other methods can be used to verify the value of $x$?
4. How would the equation change if the angle between $COD$ was not given?
5. Can we solve for any unknown angles if we had additional information about angle $C$ or angle $D$?

Tip: In geometry, always remember that angles on a straight line sum to $180^\circ$, and this property is often useful for solving problems involving linear arrangements of points.