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Using Properties of Perpendicular Lines

Using Properties of Perpendicular Lines 1.4 - Solution

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We are given information about lines in a plane.

  • Line aa is parallel to line b.b.
  • Line bb is parallel to line c.c.
  • Line dd is perpendicular to line a.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 dd intersects lines a,a, b,b, and c,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. If two lines are parallel to the same line,then they are parallel to each other.\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 aa and cc are both parallel to b.b. Therefore, according to the above theorem, lines aa and cc are parallel. a and b are parallel  and b and c are parallel 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.