In the image, we're asked to find the value of $x$ using the angle relationships in the diagram. Here’s a breakdown of how to approach this problem:

1. Identify the angle relationship: We see that angles $\angle PTQ$ and $\angle RTS$ are on a straight line and form a linear pair. Therefore, they are supplementary angles. This means: $\angle PTQ + \angle RTS = 180^\circ$

2. Set up the equation: According to the diagram:

   • $\angle PTQ = 41^\circ$
   • $\angle RTS = (2x - 1)^\circ$

   Substitute these values into the supplementary angle equation: $41 + (2x - 1) = 180$

3. Solve for $x$:

   • Simplify the equation: $41 - 1 + 2x = 180$ $40 + 2x = 180$
   • Subtract 40 from both sides: $2x = 140$
   • Divide by 2: $x = 70$

Therefore, the value of $x$ is 70.

Would you like a more detailed breakdown, or do you have any questions?

Here are five questions to deepen your understanding:

1. What other angle relationships, besides supplementary angles, could help solve similar problems?
2. How would you approach this problem if additional angles or variables were involved?
3. What if the two angles didn’t form a linear pair but were vertically opposite—how would that affect the solution?
4. How can you verify the value of $x$ after solving it in angle-based problems?
5. In what other real-life contexts might understanding supplementary angles be useful?

Tip: Always check if angles in a diagram form a linear pair (straight line) or are vertically opposite, as this can simplify the calculations.