A triangle ABC has vertices on the grid lines. The center of the inscribed circle O is also on a grid intersection. Determine the possible values for the angle AOK, given that angle ABC is β.

Explanation:

This problem involves geometry and requires calculating angles within a triangle and its inscribed circle. The solution will involve using properties of inscribed circles and angles.

Possible values for angle AOK:

The problem provides a table with values for angle ABC (β) and corresponding values for angle AOK. We need to fill in the missing values.

Note: The image is missing the context to solve for the second row. Based on the formula provided for the third row, we can infer the relationship between angle ABC and angle AOK. The formula for angle AOK in terms of angle ABC (β) is typically given by: $$AOK = 180° - β$$. However, the given formula $$90° + rac{3eta}{2}$$ seems to relate to a different angle or context. Without further information or clarification on how AOK is derived from β in this specific problem, it is impossible to definitively fill the table or solve for the missing values. The provided image presents a geometric problem with a table to be filled, but the underlying geometric relationships or formulas required to solve it are not fully explicit or might be derived from a larger context not provided.