1. **Problem statement:** We need to find the size of angle $k$ given that another angle in the figure is $81^\circ$. 2. **Assumption:** Since the problem involves angles and one angle is $81^\circ$, it is likely that angles $k$ and $81^\circ$ are related by a geometric rule such as supplementary angles (sum to $180^\circ$) or complementary angles (sum to $90^\circ$). Without additional information, the most common case is that they are supplementary. 3. **Formula:** For supplementary angles, $$k + 81^\circ = 180^\circ$$ 4. **Solve for $k$:** $$k = 180^\circ - 81^\circ$$ 5. **Calculate:** $$k = 99^\circ$$ 6. **Conclusion:** The size of angle $k$ is $99^\circ$. This solution assumes the angles are supplementary, which is typical in such problems unless otherwise specified.