#1) Given; #2) Vertical angles are congruent; and #3) AA similarity theorem.
Explanation:
#1) This is a "given" for the proof. It requires no further explanation. #2) ∠ABC and ∠DBE are opposite angles that share only a vertex. This is the definition of vertical angles, and we know that vertical angles are congruent. #3) The AA similarity theorem states that if two corresponding angles of two triangles are congruent, then the triangles are similar. This is because all triangles have a sum of 180°. If two of the angles are congruent, the third angle in both would have to be the same to make them equal 180°. Thus the triangles are similar.
