https://file.aiquickdraw.com/mathbot/file/17309099972625mtajvc7.jpg

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?

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#### 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 \( 180^\circ \).

Find the value of x + y in Diagram 16, given that PQT and RQU are straight lines, and ∠TQU = 65°.

Solve a geometry problem involving straight lines and supplementary angles. Given PQT and RQU as straight lines, and ∠TQU = 65°, find the value of x + y using angle properties. Suitable for Grades 7-9 students.

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