Question

ab is parallel to cd and perpendicular to both ef and gh. which statements are true? select two correct answers. a cd is parallel to gh. b ab is parallel to gh. c ef is parallel to gh. d ef is perpendicular to cd. e ef is perpendicular to gh.

Explanation:

Step1: Recall parallel - perpendicular vector relationships

Given $\overrightarrow{AB}\parallel\overrightarrow{CD}$ and $\overrightarrow{AB}\perp\overrightarrow{EF}$, $\overrightarrow{AB}\perp\overrightarrow{GH}$. Since $\overrightarrow{AB}\parallel\overrightarrow{CD}$, vectors $\overrightarrow{CD}$ has the same perpendicular - parallel relationships as $\overrightarrow{AB}$.

Step2: Analyze each option

• Option A: $\overrightarrow{CD}$ is parallel to $\overrightarrow{GH}$. There is no information to suggest this. $\overrightarrow{AB}$ is perpendicular to $\overrightarrow{GH}$ and parallel to $\overrightarrow{CD}$, so $\overrightarrow{CD}\perp\overrightarrow{GH}$, A is false.
• Option B: $\overrightarrow{AB}$ is parallel to $\overrightarrow{GH}$. Given $\overrightarrow{AB}\perp\overrightarrow{GH}$, B is false.
• Option C: $\overrightarrow{EF}$ is parallel to $\overrightarrow{GH}$. There is no information to suggest this. They could be at any non - parallel non - perpendicular angle, C is false.
• Option D: Since $\overrightarrow{AB}\parallel\overrightarrow{CD}$ and $\overrightarrow{AB}\perp\overrightarrow{EF}$, then $\overrightarrow{EF}\perp\overrightarrow{CD}$ because if two vectors are parallel, a vector perpendicular to one is perpendicular to the other.
• Option E: There is no information to suggest $\overrightarrow{EF}\perp\overrightarrow{GH}$. They could be at any non - parallel non - perpendicular angle, E is false.

Answer:

D. $\overrightarrow{EF}$ is perpendicular to $\overrightarrow{CD}$