1. **Problem statement:** We have a triangle with two sides of lengths 19 and 20. We want to find the smallest possible whole-number length for the third side. 2. **Formula and rule:** For any triangle with sides $a$, $b$, and $c$, the triangle inequality must hold: $$ |a - b| < c < a + b $$ This means the length of any side must be greater than the difference of the other two sides and less than their sum. 3. **Apply the triangle inequality:** Given sides 19 and 20, let the third side be $x$. We have: $$ |19 - 20| < x < 19 + 20 $$ $$ 1 < x < 39 $$ 4. **Find the smallest whole number:** Since $x$ must be greater than 1, the smallest whole number length for the third side is: $$ \boxed{2} $$ This ensures the three sides can form a valid triangle.