1. **Stating the problem:** We need to find the angle $j$ in the given polygon with multiple interior and exterior angles. 2. **Understanding the problem:** The polygon has several interior and exterior angles given, including $i=125^\circ$, exterior angles $78^\circ$, $60^\circ$, $78^\circ$, $102^\circ$, and interior angles $55^\circ$, and unknown angles $j$ and $k$. 3. **Key rule:** The sum of the exterior angles of any polygon is always $360^\circ$. 4. **Sum of given exterior angles:** $$78^\circ + 60^\circ + 78^\circ + 102^\circ + j = 360^\circ$$ 5. **Calculate $j$:** $$j = 360^\circ - (78^\circ + 60^\circ + 78^\circ + 102^\circ)$$ $$j = 360^\circ - 318^\circ$$ $$j = 42^\circ$$ 6. **Answer:** The angle $j$ is $42^\circ$. This uses the fundamental property of polygons that the sum of exterior angles is always $360^\circ$, allowing us to find the missing exterior angle $j$ by subtracting the sum of the known exterior angles from $360^\circ$.