1. **Stating the problem:** We need to find the size of angle $e$ given that it is adjacent to a $78^\circ$ angle and there is a right angle ($90^\circ$) formed by a vertical dashed line and the lower black solid line. 2. **Understanding the setup:** The right angle ($90^\circ$) is split into two angles: one is $78^\circ$ and the other is angle $e$. 3. **Formula used:** The sum of angles on a straight line or around a point is $180^\circ$, but here since the right angle is $90^\circ$, the sum of the two adjacent angles must be $90^\circ$. 4. **Calculation:** $$ e + 78^\circ = 90^\circ $$ 5. **Solving for $e$:** $$ e = 90^\circ - 78^\circ = 12^\circ $$ 6. **Conclusion:** The size of angle $e$ is $12^\circ$.