Question

#1: ∠egf and ∠bgc are vertical angles. *
false
#2: ∠bgc and ∠cgd are complementary angles. *
choose
#3: ∠bga and ∠cgd are adjacent

Explanation:

Step1: Recall vertical - angle definition

Vertical angles are formed by two intersecting lines and are opposite each other. $\angle EGF$ and $\angle BGC$ are not formed by the same two - intersecting lines, so the statement “$\angle EGF$ and $\angle BGC$ are vertical angles” is false.

Step2: Recall complementary - angle definition

Complementary angles add up to $90^{\circ}$. From the figure, if $\angle DGE = 90^{\circ}$, and $\angle BGC+\angle CGD+\angle DGE+\angle EGF = 360^{\circ}$ (a full - circle around point $G$), and there is no information indicating that $\angle BGC+\angle CGD = 90^{\circ}$, we cannot say they are complementary. But if we assume that the lines are arranged such that $\angle BGC+\angle CGD = 90^{\circ}$, they are complementary. Without more information, we assume the general case. Since $\angle DGE = 90^{\circ}$ and we have no indication of a right - angle split between $\angle BGC$ and $\angle CGD$, the answer is false.

Step3: Recall adjacent - angle definition

Adjacent angles have a common side and a common vertex and no common interior points. $\angle BGA$ and $\angle CGD$ do not have a common side, so they are not adjacent angles.

Answer:

#1: false
#2: false
#3: false