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Thread: Check my proof about perpendicular lines which are parallel

  1. #1
    Senior Member OReilly's Avatar
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    Check my proof about perpendicular lines which are parallel

    I have to prove this theorem:

    If two different lines $\displaystyle a$ and $\displaystyle b$ are perpendicular to plane $\displaystyle \alpha$ then $\displaystyle a$ and $\displaystyle b$ are parallel lines.

    My proof:
    Lines $\displaystyle a$ and $\displaystyle b$ are perpendicular on plane $\displaystyle \alpha$ in points A and B.

    Through points A and B there is line $\displaystyle c$ which is perpendicular to lines $\displaystyle a$ and $\displaystyle b$.
    Lines $\displaystyle b$ and $\displaystyle c$ form plane $\displaystyle \beta$ perpendicular to line $\displaystyle d$ (in point B).
    Through point B there is line $\displaystyle e$ which intersects line $\displaystyle a$ and is normal to line $\displaystyle d$.
    Lines $\displaystyle e$ and $\displaystyle b$ form plane $\displaystyle \gamma$ which is normal to line $\displaystyle d$, but then there would exist two planes that contains point B and are perpendicular to line $\displaystyle d$ which is possible only if $\displaystyle \beta=\gamma$.
    Then plane $\displaystyle \beta$ would contain also lines $\displaystyle c$ and $\displaystyle e$ and also line $\displaystyle a$ (because lines $\displaystyle c$ and $\displaystyle e$ intersects line $\displaystyle a$) which means that lines $\displaystyle a$ and $\displaystyle b$ are both in plane $\displaystyle \beta$.

    Lines $\displaystyle a$ and $\displaystyle b$ don't intersect because than it would exist two lines perpendicular to line $\displaystyle c$ in one point (intersection point) so they are parallel.


    Is my proof ok?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by OReilly View Post
    I have to prove this theorem:

    If two different lines $\displaystyle a$ and $\displaystyle b$ are perpendicular to plane $\displaystyle \alpha$ then $\displaystyle a$ and $\displaystyle b$ are parallel lines.

    My proof:
    Lines $\displaystyle a$ and $\displaystyle b$ are perpendicular on plane $\displaystyle \alpha$ in points A and B.

    Through points A and B there is line $\displaystyle c$ which is perpendicular to lines $\displaystyle a$ and $\displaystyle b$.
    Lines $\displaystyle b$ and $\displaystyle c$ form plane $\displaystyle \beta$ perpendicular to line $\displaystyle d$ (in point B).
    Through point B there is line $\displaystyle e$ which intersects line $\displaystyle a$ and is normal to line $\displaystyle d$.
    Lines $\displaystyle e$ and $\displaystyle b$ form plane $\displaystyle \gamma$ which is normal to line $\displaystyle d$, but then there would exist two planes that contains point B and are perpendicular to line $\displaystyle d$ which is possible only if $\displaystyle \beta=\gamma$.
    Then plane $\displaystyle \beta$ would contain also lines $\displaystyle c$ and $\displaystyle e$ and also line $\displaystyle a$ (because lines $\displaystyle c$ and $\displaystyle e$ intersects line $\displaystyle a$) which means that lines $\displaystyle a$ and $\displaystyle b$ are both in plane $\displaystyle \beta$.

    Lines $\displaystyle a$ and $\displaystyle b$ don't intersect because than it would exist two lines perpendicular to line $\displaystyle c$ in one point (intersection point) so they are parallel.


    Is my proof ok?
    Looks okay to me, but if I may just say that reading the title of this post made my morning!

    -Dan
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