In this image, we are asked to find the missing angles $x$ and $y$. Here is how we can approach the problem step by step:

1. The angle marked with a small square indicates a right angle, which is $90^\circ$.
2. The line extending vertically indicates that the sum of the angles on that line must be $180^\circ$, since they form a straight line.
3. The given angle is $38^\circ$, so we can calculate the adjacent angle $x$ because together they form a straight line: $x = 180^\circ - 38^\circ = 142^\circ.$
4. Now, to find $y$, note that $y$ is the complementary angle to the right angle (because $y$ and the right angle are on the same side of the intersection), so: $y = 90^\circ - 38^\circ = 52^\circ.$

Therefore, the missing angles are:

• $x = 142^\circ$,
• $y = 52^\circ$.

Would you like more details or have any further questions?

Here are 5 related questions for further practice:

1. What is the sum of all angles around a point?
2. How would you find a missing angle in a triangle given two known angles?
3. How does the concept of vertical angles help in solving problems with intersecting lines?
4. What is the relationship between complementary and supplementary angles?
5. How do you apply angle properties to parallel lines cut by a transversal?

Tip: Remember, angles on a straight line always sum up to $180^\circ$.