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The diagram provided shows a geometrical figure involving angles at points \( A \), \( P \), and \( Q \). We need to determine the values of \( x \) and \( y \), which are angles at point \( P \).

### Step-by-step Solution:

1. **Angle \( x \):**
   - Angle \( x \) is part of a straight line formed by points \( A \), \( P \), and \( Q \).
   - The straight angle at \( A \) is 180°, and angle \( x \) plus the given angle of 78° must sum up to 180° (since they are supplementary angles on a straight line).
   
   \[
   x + 78^\circ = 180^\circ
   \]
   Solving for \( x \):
   \[
   x = 180^\circ - 78^\circ = 102^\circ
   \]

2. **Angle \( y \):**
   - Angle \( y \) is at point \( P \), and the diagram indicates it is the external angle of a circle or near a curved edge.
   - Since the angle \( y \) forms part of a right angle (90°) next to angle \( x \), we can deduce that:
   
   \[
   x + y = 180^\circ \quad (\text{since they are angles on a straight line around point } P)
   \]
   With \( x = 102^\circ \), we get:
   \[
   102^\circ + y = 180^\circ
   \]
   Solving for \( y \):
   \[
   y = 180^\circ - 102^\circ = 78^\circ
   \]

### Final Answer:
- \( x = 102^\circ \)
- \( y = 78^\circ \)

### Reasons:
- \( x \) is found using the property of supplementary angles on a straight line.
- \( y \) is found using the fact that angles on a straight line sum to 180°.

Would you like more details or have any questions?

Here are five follow-up questions based on this:
1. Why do supplementary angles on a straight line sum to 180°?
2. How would the situation change if the angle at \( A \) was different?
3. What are the properties of exterior angles in geometric shapes like triangles?
4. Can angle \( y \) be greater than 90°? Why or why not?
5. How do we handle angles in circles if this diagram represents a part of a circular arc?

**Tip:** Always remember that angles on a straight line sum to 180°, and this is useful for solving problems involving linear geometry.

Determine the sizes of x and y. Give reasons.

Learn how to determine the sizes of angles x and y using the concept of supplementary angles on a straight line. This problem is suitable for students in Grades 6-8 and covers key concepts in geometry.

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