To solve this problem, we are dealing with a right triangle and the properties of its angles.

Steps to Solve:

1. Understand the Angle Relationships in a Triangle:

   • The sum of the angles in any triangle is $180^\circ$.
   • In a right triangle, one of the angles is $90^\circ$, leaving the other two angles to sum to $90^\circ$.

2. Label the Angles:

   • Given: $y = 38^\circ$, $67^\circ$, and $x$ as the remaining unknown angle.

Solution for $x$:

The sum of the angles in a triangle is $180^\circ$: $x + y + 90^\circ = 180^\circ$ Substitute $y = 38^\circ$: $x + 38^\circ + 90^\circ = 180^\circ$ $x = 180^\circ - 128^\circ$ $x = 52^\circ$

Solution for $y$:

From the diagram, $y$ is already labeled as $38^\circ$.

Final Answer:

• $x = 52^\circ$
• $y = 38^\circ$

Would you like further explanation, or do you have other questions?

Related Questions:

1. How can you confirm the sum of angles in any triangle?
2. What are the properties of right triangles that make them unique?
3. How would the solution change if the triangle were not a right triangle?
4. Can you explain why angle $x$ is complementary to $y$?
5. How do trigonometric ratios relate to this triangle’s angles?

Tip: Always verify the angle sum when working with triangles—it ensures accuracy.