Two angles are said to be adjacent when they share a common vertex and side.
Let's analyze each option.
∠CED and ∠DEA share a common vertex (point E) and they are both on the same side of the line CG, thus they are adjacent.
∠AEB and ∠DEA also share a common vertex (point E) and they are both on the same side of the line BD. These two angles are adjacent
∠HAE and ∠DEA. These angles don't have a common vertex. ∠HAE's vertex is at point A and ∠DEA's vertex is at point E, thus the angles are not adjacent.
∠EAF and ∠HAE. These angles have a common vertex (point A) and both are above the line HF, thus they are adjacent.
The correct choices are 1, 2, and 4