To solve for $x$, we can use the fact that the angles along a straight line add up to $180^\circ$. This means the sum of the two given angles must equal $180^\circ$.

We have two angles:

• One is $34^\circ$,
• The other is $(1 + 5x)^\circ$.

Thus, the equation becomes: $34^\circ + (1 + 5x)^\circ = 180^\circ$

Let's solve for $x$:

$34 + 1 + 5x = 180$ $35 + 5x = 180$ $5x = 180 - 35$ $5x = 145$ $x = \frac{145}{5}$ $x = 29$

Therefore, the value of $x$ is $29$.

Would you like more details or have any further questions?

Here are five related questions to expand on this:

1. How do you solve for unknown angles in a triangle?
2. What is the sum of the interior angles of a triangle?
3. How can you use the exterior angle theorem to solve for $x$?
4. How does the concept of supplementary angles apply to straight lines?
5. How would the solution change if the second angle was in terms of another variable?

Tip: Always remember that angles on a straight line are supplementary, meaning they sum up to $180^\circ$.