If a|| b, a is perpendicular to c and b is perpendicular to d how do I prove c|| d ?

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valentin68 | College Teacher | (Level 3) Associate Educator

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We suppose all lines lie in the same plane. From the given data we have

`a||b and b _|_ d rArr a _|_ d ` ` `

Also from the text data we have `a _|_ c` . Since both `c and d` lines are perpendicular to `a` , they need to be parallel one to the other if they lie in the same plane. (The condition `c` and `d` are in the same plane need to be imposed otherwise the lines could just not intersect in space without being parallel. This case is illustrated in the figure below). Therefore `d ||c` .

Another way to do this is to assign each of the lines a slope `s` and consider the general equation of a line in a plane of the form

`y =s*x + n`

`a||b ` means both lines `a and b` have the same slope `s =m`

`b _|_d` means line `d` has slope `s= -1/m`

`a_|_c` means line `c` has slope `s=-1/m`

Since lines `d and c` have the same slope `-1/m` they are parallel to each other `d||c` .

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