1. **State the problem:** We need to find the size of angle $k$ given a diagram with intersecting lines and angles $86^\circ$, $78^\circ$, and $102^\circ$. 2. **Recall the rule:** Adjacent angles on a straight line (linear pair) sum to $180^\circ$. 3. **Apply the rule to angles $86^\circ$ and $k$:** Since they are adjacent and form a straight line, $$86 + k = 180$$ 4. **Solve for $k$:** $$k = 180 - 86 = 94$$ 5. **Check the other pair:** Angles $78^\circ$ and $102^\circ$ also form a linear pair, $$78 + 102 = 180$$ which confirms the lines are straight and the rule applies. **Final answer:** $$\boxed{94^\circ}$$