The triangle $K_2$ has as its vertices the feet of the altitudes of a non-right triangle $K_1$. Find all possibilities for the sizes of the angles of $K_1$ for which the triangles $K_1$ and $K_2$ are similar.
To solve the problem, we need to determine the conditions under which the triangle \( K_1 \) and its orthic triangle \( K_2 \) are similar. The orthic triangle \( K_2 \) is formed by the feet of the altitudes of \( K_1 \).

1. **Identify the angles of the orthic triangle \( K_2 \):**
   - Let \( A, B, \) and \( C \) be the angles of triangle \( K_1 \).
   - The angles of the orthic triangle \( K_2 \) are given by \( 180^\circ - 2A \), \( 180^\circ - 2B \), and \( 180^\circ - 2C \).

2. **Set up the similarity condition:**
   - For \( K_1 \) and \( K_2 \) to be similar, their corresponding angles must be proportional. This means we need:
     \[
     A = 180^\circ - 2A, \quad B = 180^\circ - 2B, \quad C = 180^\circ - 2C
     \]

3. **Solve the angle equations:**
   - Consider the equation \( A = 180^\circ - 2A \):
     \[
     A + 2A = 180^\circ
     \]
     \[
     3A = 180^\circ
     \]
     \[
     A = 60^\circ
     \]
   - Similarly, solving for \( B \) and \( C \):
     \[
     B = 180^\circ - 2B \implies 3B = 180^\circ \implies B = 60^\circ
     \]
     \[
     C = 180^\circ - 2C \implies 3C = 180^\circ \implies C = 60^\circ
     \]

4. **Verify the solution:**
   - If \( A = 60^\circ \), \( B = 60^\circ \), and \( C = 60^\circ \), then triangle \( K_1 \) is an equilateral triangle.
   - The orthic triangle of an equilateral triangle is also an equilateral triangle, hence \( K_1 \) and \( K_2 \) are similar.

5. **Consider other possibilities:**
   - Since the only solution to the equations \( A = 180^\circ - 2A \), \( B = 180^\circ - 2B \), and \( C = 180^\circ - 2C \) is \( A = B = C = 60^\circ \), there are no other possibilities.

Conclusion:
The only possibility for the angles of \( K_1 \) such that \( K_1 \) and \( K_2 \) are similar is when \( K_1 \) is an equilateral triangle.

The final answer is \( \boxed{ A = 60^\circ, B = 60^\circ, C = 60^\circ } \).