1. **State the problem:** We need to find the size of angle $k$ in two triangles that share a common side, with given angle $81^\circ$ and pairs of equal sides. 2. **Identify the key property:** Since the triangles have three pairs of equal sides, they are congruent by the SSS (Side-Side-Side) criterion. 3. **Apply congruence:** Congruent triangles have corresponding angles equal. Therefore, the angle corresponding to the $81^\circ$ angle in the other triangle is also $81^\circ$. 4. **Use the fact that angles on a straight line sum to $180^\circ$:** The angles $81^\circ$ and $k$ are adjacent and form a straight line, so $$k + 81^\circ = 180^\circ$$ 5. **Solve for $k$:** $$k = 180^\circ - 81^\circ = 99^\circ$$ **Final answer:** $$\boxed{99^\circ}$$