Question

in the cube shown below, which lines are parallel? select all that apply. ad and bc ab and ef ab and dh ae and cg

Explanation:

Step1: Recall parallel - line property in cube

In a cube, parallel lines are lines that never intersect and are in the same or parallel planes.

Step2: Analyze each pair

• For $\overrightarrow{AD}$ and $\overrightarrow{BC}$: They are on the same face (the front - face ABCD), and by the definition of a cube, opposite sides of a square face are parallel.
• For $\overrightarrow{AB}$ and $\overrightarrow{EF}$: $\overrightarrow{AB}$ is on the front - face ABCD and $\overrightarrow{EF}$ is on the bottom - face EFGH. Since the front - face and the bottom - face are parallel planes, and $\overrightarrow{AB}$ and $\overrightarrow{EF}$ have the same direction, they are parallel.
• For $\overrightarrow{AB}$ and $\overrightarrow{DH}$: $\overrightarrow{AB}$ is on the front - face ABCD and $\overrightarrow{DH}$ is on the side - face ADHE. They are not parallel as they are not in the same or parallel planes and will intersect if extended.
• For $\overrightarrow{AE}$ and $\overrightarrow{CG}$: $\overrightarrow{AE}$ is on the side - face ADHE and $\overrightarrow{CG}$ is on the side - face BCGF. Since the side - faces ADHE and BCGF are parallel planes, and $\overrightarrow{AE}$ and $\overrightarrow{CG}$ have the same direction, they are parallel.

Answer:

$\overrightarrow{AD}$ and $\overrightarrow{BC}$, $\overrightarrow{AB}$ and $\overrightarrow{EF}$, $\overrightarrow{AE}$ and $\overrightarrow{CG}$