Using Properties of Perpendicular Lines

We are given information about lines in a plane.

• Line $a$ is parallel to line $b.$
• Line $b$ is parallel to line $c.$
• Line $d$ is perpendicular to line $a.$

We are given statements and asked which of them that must be true. Let's draw a diagram to illustrate the situation.

Since line $d$ intersects lines $a,$ $b,$ and $c,$ it is not parallel to any of them. Therefore, neither statement I, nor II can be true true. To determine whether statement III is true, we will use the following theorem. $\begin{gathered} \textit{If two lines are parallel to the same line,}\\ \textit{then they are parallel to each other.}\\[0.8em] \end{gathered}$ In our case, lines $a$ and $c$ are both parallel to $b.$ Therefore, according to the above theorem, lines $a$ and $c$ are parallel. $\begin{gathered} a \text{ and } b \text{ are parallel } \\ \text{ and } \\ b \text{ and } c \text{ are parallel } \\ \Downarrow \\ a \text{ and } c \text{ are parallel } \end{gathered}$ If two lines are parallel they can not be perpendicular. Thus, statement III is the one that must be true.