Question

construct arguments determine whether each statement is always, sometimes, or never true. justify your argument. if $overleftrightarrow{ab}$ is perpendicular to $overleftrightarrow{bc}$ and $overleftrightarrow{bc}$ is perpendicular to $overleftrightarrow{cd}$, then $overleftrightarrow{ab}$ is parallel to $overleftrightarrow{cd}$. this is true only if $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ are

Explanation:

Step1: Recall perpendicular - parallel relationship

If \(AB\perp BC\) and \(BC\perp CD\), in a two - dimensional plane, \(AB\) and \(CD\) are parallel. In three - dimensional space, \(AB\) and \(CD\) may not be parallel (they can be skew lines).

Step2: Determine the answer

Since the statement does not specify whether we are in 2 - D or 3 - D space, the statement "If \(AB\) is perpendicular to \(BC\) and \(BC\) is perpendicular to \(CD\), then \(AB\) is parallel to \(CD\)" is sometimes true.

Answer:

Sometimes. Justification: In a two - dimensional plane, two lines perpendicular to the same line are parallel. But in three - dimensional space, two lines perpendicular to the same line may be skew (not parallel and not intersecting).