Question

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Answer 1

Answer:

Option 4:  A ║B

Step-by-step explanation:

Given the following statements:

$\displaystyle\mathsf{\overline{X}}$ is ⊥ $\displaystyle\mathsf{\overline{A}}$

$\displaystyle\mathsf{\overline{X}}$ is ⊥ $\displaystyle\mathsf{\overline{B}}$

We can apply the Lines Perpendicular to a Transversal Theorem, which states that if two lines intersect and are perpendicular to that same transversal, then those two lines are parallel.  

In the given conditional statement, $\displaystyle\mathsf{\overline{X}}$ ⊥ $\displaystyle\mathsf{\overline{A}}$  and  $\displaystyle\mathsf{\overline{X}}$ ⊥ $\displaystyle\mathsf{\overline{B}}$, line $\displaystyle\mathsf{\overline{X}}$ is the transversal that cuts through both lines  $\displaystyle\mathsf{\overline{A}}$ and $\displaystyle\mathsf{\overline{B}}$.  

Therefore, lines  $\displaystyle\mathsf{\overline{A}}$ and $\displaystyle\mathsf{\overline{B}}$ are perpendicular to line $\displaystyle\mathsf{\overline{X}}$.  Hence, by the Lines Perpendicular to a Transversal Theorem,  $\displaystyle\mathsf{\overline{A}}$ ║ $\displaystyle\mathsf{\overline{B}}$.