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.