https://file.aiquickdraw.com/mathbot/file/17261656100222r0pf45o.jpg

Let's analyze the problem step by step.

### Given Information:
- \( QT \perp PR \) (QT is perpendicular to PR, meaning \( \angle QTR = 90^\circ \)).
- \( \angle TQR = 40^\circ \).
- \( \angle SPR = 30^\circ \).

You are tasked with finding the angles \( x \) and \( y \) marked in the diagram.

### Step-by-Step Solution:

1. **Angle in a Triangle (TQR)**:
   In triangle \( TQR \), the sum of interior angles must equal \( 180^\circ \).
   \[
   \angle TQR + \angle QTR + \angle QRT = 180^\circ
   \]
   Substitute the given values:
   \[
   40^\circ + 90^\circ + \angle QRT = 180^\circ
   \]
   Solving for \( \angle QRT \):
   \[
   \angle QRT = 180^\circ - 130^\circ = 50^\circ
   \]
   So, \( \angle QRT = 50^\circ \).

2. **Angle SPR**:
   From the diagram, \( \angle SPR = 30^\circ \) is given, which corresponds to the marked angle \( y \). Hence:
   \[
   y = 30^\circ
   \]

3. **Angle x** (Exterior Angle Theorem):
   Using the exterior angle theorem, the exterior angle \( x \) at point \( P \) is equal to the sum of the two opposite interior angles in triangle \( TQR \):
   \[
   x = \angle TQR + \angle QRT = 40^\circ + 50^\circ = 90^\circ
   \]

Thus, the values of \( x \) and \( y \) are:
\[
x = 90^\circ, \quad y = 30^\circ
\]

### Final Answer:
- \( x = 90^\circ \)
- \( y = 30^\circ \)

Would you like further clarification on any part of the solution?

### Follow-up Questions:
1. How can the exterior angle theorem be applied to different triangle problems?
2. What are the different properties of perpendicular lines in triangles?
3. How do we calculate angles when one is given as a perpendicular?
4. Why is the sum of the interior angles of a triangle always \( 180^\circ \)?
5. How does knowing one angle help in determining other angles in a geometric figure?

### Tip:
Always look for right angles in geometric figures as they simplify calculations by providing clear angle relationships.

In the figure given, QT ⊥ PR, ∠TQR = 40° and ∠SPR = 30°. Find x and y.

This problem involves solving for angles x and y in a triangle with a perpendicular line, using the sum of angles in a triangle and the exterior angle theorem. Suitable for students in Grades 8-10.

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