1. **State the problem:** We have a large triangle with angles labeled $n$ at the bottom-left vertex, $3n$ at the top vertex, and the base angle split into two adjacent angles $2m$ and $3m$ on a straight horizontal line. 2. **Analyze the angles:** The large triangle's angles must sum to $180^\circ$. The angles are $n$, $3n$, and the angle at the bottom-right vertex, which is split into $2m$ and $3m$. 3. **Sum of angles on a straight line:** Since $2m$ and $3m$ are adjacent angles on a straight line, their sum is $180^\circ$: $$2m + 3m = 5m = 180^\circ$$ 4. **Sum of angles in the triangle:** The three angles of the triangle are $n$, $3n$, and the angle at the bottom-right vertex, which is $5m$ (since $2m + 3m = 5m$). So: $$n + 3n + 5m = 180^\circ$$ $$4n + 5m = 180^\circ$$ 5. **Relate $m$ and $n$ using the vertical line:** The vertical line creates two triangles. The top vertex angle is $3n$, and the bottom-left angle is $n$. The vertical line is perpendicular to the base, so the angles $2m$ and $3m$ are complementary to the angles in the smaller triangle. 6. **Use the fact that the vertical line is perpendicular:** The angles $2m$ and $3m$ are adjacent and sum to $180^\circ$, so the vertical line is perpendicular to the base, making the angle between the vertical line and the base $90^\circ$. 7. **Use the triangle angle sum in the smaller triangle:** The smaller triangle has angles $3n$ (top), $90^\circ$ (right angle at the base), and the remaining angle at the bottom-left vertex is $n$. 8. **Sum of angles in the smaller triangle:** $$3n + 90^\circ + n = 180^\circ$$ $$4n + 90^\circ = 180^\circ$$ $$4n = 90^\circ$$ $$n = 22.5^\circ$$ **Final answer:** $$\boxed{n = 22.5}$$