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if a||b, b||c, and c is perpendicular to d, how is line a related to line d

We are given that a is parallel to b, b is parallel to c and c is perpendicular to d and we are asked to determine how line a is related to line d.


There are two possibilities:


(1) If all of the lines are in the same plane then a is perpendicular to d. It is always true that 2 lines parallel to a third line are parallel; thus since a and c are parallel to b then a is parallel to c. In a plane, parallel lines form congruent corresponding angles with a given transversal. Since c is perpendicular to d the angles formed are right angles so a forms right angles with d and is perpendicular to d.


(2) If it is possible that the lines are not all coplanar then a and d could be skew. (Skew lines are noncoplanar lines that do not intersect.) 


As asked, the answer is that the relationship cannot be determined.

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