Question

#13: determine which statements are true. select all that apply.
∠hkg and ∠gkf are adjacent angles.
∠dke and ∠jkh are adjacent angles.
∠gkf and ∠dke are vertical angles.
∠jkh and ∠ekf are vertical angles.
∠dke and ∠ekf are complementary angles.
∠jkd and ∠dkf are complementary angles.
∠hkg and ∠gke are supplementary angles
∠dke and ∠ekf are supplementary angles

Explanation:

Step1: Recall adjacent - angles definition

Adjacent angles share a common vertex and a common side. $\angle HKG$ and $\angle GKF$ share vertex $K$ and side $\overrightarrow{KG}$, so they are adjacent angles. $\angle DKE$ and $\angle JKH$ do not share a common side, so they are not adjacent angles.

Step2: Recall vertical - angles definition

Vertical angles are non - adjacent angles formed by two intersecting lines. $\angle GKF$ and $\angle DKE$ are not formed by two intersecting lines, so they are not vertical angles. $\angle JKH$ and $\angle EKF$ are vertical angles as they are non - adjacent angles formed by the intersection of lines containing $\overrightarrow{JH}$ and $\overrightarrow{EF}$.

Step3: Recall complementary - angles definition

Complementary angles add up to $90^{\circ}$. $\angle DKE$ and $\angle EKF$ do not necessarily add up to $90^{\circ}$. $\angle JKD$ and $\angle DKF$ are complementary since $\angle JKF = 90^{\circ}$.

Step4: Recall supplementary - angles definition

Supplementary angles add up to $180^{\circ}$. $\angle HKG$ and $\angle GKE$ do not necessarily add up to $180^{\circ}$. $\angle DKE$ and $\angle EKF$ are supplementary since they form a linear pair.

Answer:

$\angle HKG$ and $\angle GKF$ are adjacent angles; $\angle JKH$ and $\angle EKF$ are vertical angles; $\angle JKD$ and $\angle DKF$ are complementary angles; $\angle DKE$ and $\angle EKF$ are supplementary angles.