Check my proof about perpendicular lines which are parallel

**OReilly** · January 18th 2007, 05:13 AM

I have to prove this theorem:

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

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

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

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

Is my proof ok?

**topsquark** · January 18th 2007, 05:18 AM

Originally Posted by OReilly

I have to prove this theorem:

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

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

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

Lines $a$ and $b$ don't intersect because than it would exist two lines perpendicular to line $c$ 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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