There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible?

[i]Proposed by E. Bakaev[/i]
1. **Understanding the problem**: We need to find the maximum number of points \( N \) on a plane such that any three points form a triangle with angles that are all natural numbers (in degrees).

2. **Analyzing the constraints**: For any triangle, the sum of the angles is \( 180^\circ \). Since the angles must be natural numbers, each angle must be an integer between \( 1^\circ \) and \( 179^\circ \).

3. **Convex hull consideration**: Consider a point \( A \) on the convex hull of the set of points, and let \( B \) and \( C \) be the points adjacent to \( A \) on the convex hull. The angle \( \angle BAC \) must be an integer.

4. **Bounding the number of points**: If there are \( N \) points, then there are \( N-3 \) points inside the triangle \( \triangle ABC \). For \( \angle BAC \) to be an integer and for the triangle to have integer angles, the sum of the angles at \( A \), \( B \), and \( C \) must be \( 180^\circ \).

5. **Maximum angle consideration**: If there are at least \( 178 \) other points inside \( \angle BAC \), then \( \angle BAC \geq 179^\circ \). This is impossible because the sum of the angles in any triangle is \( 180^\circ \), and having one angle \( \geq 179^\circ \) would leave less than \( 1^\circ \) for the other two angles combined, which is not possible.

6. **Constructing the maximum number of points**: To construct the maximum number of points, consider placing \( 180 \) equidistant points on a circle. This ensures that any triangle formed by these points will have angles that are natural numbers. 

7. **Verification**: With \( 180 \) points on a circle, the angles between any three points will be multiples of \( 1^\circ \), ensuring that all angles in any triangle formed are natural numbers.

Conclusion:
The maximum number of points \( N \) for which any three points form a triangle with natural number angles is \( 180 \).

The final answer is \( \boxed{180} \)