1. **State the problem:** We need to find the size of angle $k$ in a triangle where one angle is $81^\circ$ and the triangle has two pairs of equal sides, indicating it is isosceles. 2. **Recall the properties of isosceles triangles:** In an isosceles triangle, angles opposite equal sides are equal. 3. **Analyze the triangle:** Since the triangle has two pairs of equal sides, it is actually an isosceles triangle with two equal angles adjacent to angle $k$ and the $81^\circ$ angle. 4. **Set variables:** Let the two equal angles adjacent to $k$ be $x$ each. 5. **Use the triangle angle sum formula:** The sum of angles in any triangle is $180^\circ$. $$k + 81 + 2x = 180$$ 6. **Use the isosceles property:** The two equal sides opposite the equal angles mean those angles are equal, so $x = k$. 7. **Substitute $x = k$ into the equation:** $$k + 81 + 2k = 180$$ 8. **Simplify:** $$3k + 81 = 180$$ 9. **Isolate $k$:** $$3k = 180 - 81$$ $$3k = 99$$ 10. **Divide both sides by 3:** $$k = \frac{\cancel{3}99}{\cancel{3}3} = 33$$ 11. **Final answer:** The size of angle $k$ is $33^\circ$.