You have a pencil, paper and an angle of $19$ degrees made out of two equal very thin sticks. Can you construct an angle of $1$ degree using only these tools?
1. We start with an angle of $19^\circ$ made out of two equal very thin sticks.
2. To construct an angle of $1^\circ$, we need to find a way to use the $19^\circ$ angle repeatedly.
3. Notice that $361^\circ$ is equivalent to $1^\circ$ because $361^\circ = 360^\circ + 1^\circ$.
4. Therefore, if we can construct a $361^\circ$ angle, we effectively have a $1^\circ$ angle.
5. To construct a $361^\circ$ angle using the $19^\circ$ angle, we need to add the $19^\circ$ angle $19$ times:
   \[
   19 \times 19^\circ = 361^\circ
   \]
6. Since $361^\circ$ is equivalent to $1^\circ$, we have successfully constructed a $1^\circ$ angle using the $19^\circ$ angle.

The final answer is $\boxed{1^\circ}$