To solve this problem, let’s analyze the information provided in the diagram and apply angle properties accordingly.

Given:

1. $PQT$ and $RQU$ are straight lines.
2. $\angle TQU = 65^\circ$.
3. We are tasked with finding the value of $x + y$.

Solution:

Since $PQT$ and $RQU$ are straight lines, the angles around point $Q$ form a straight angle, which totals to $180^\circ$.

1. Using the straight line property at $Q$:

   • Since $TQU$ is a straight line, we know: $\angle PQS + \angle TQU = 180^\circ$
   • Substitute $\angle TQU = 65^\circ$: $x + 65^\circ = 180^\circ$
   • Solving for $x$: $x = 180^\circ - 65^\circ = 115^\circ$

2. Using the straight line property at $S$:

   • Similarly, since $PQT$ is a straight line, we can apply the same reasoning: $\angle RQS + \angle PQS = 180^\circ$
   • Substitute $x = 115^\circ$: $y + 115^\circ = 180^\circ$
   • Solving for $y$: $y = 180^\circ - 115^\circ = 65^\circ$

3. Finding $x + y$:

   • Now that we have $x = 115^\circ$ and $y = 65^\circ$, we can calculate: $x + y = 115^\circ + 65^\circ = 180^\circ$

Final Answer:

The value of $x + y$ is $180^\circ$.

Would you like further details on any step?

Additional Questions:

1. What is the property of angles on a straight line that was used here?
2. How would the solution change if $\angle TQU$ was a different angle?
3. Can you explain why the sum of angles around a point is $360^\circ$?
4. What would happen if $PQT$ and $RQU$ were not straight lines?
5. How do properties of supplementary angles help in solving such problems?

Tip:

When dealing with angles in geometric shapes, remember that the sum of angles around a straight line is always $180^\circ$.