Proof: By Euclid
(related to Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel)
- For let them meet the reference plane at points $B$ and $D$ (respectively).
- And let the straight line $BD$ have been joined.
- And let $DE$ have been drawn at right angles to $BD$ in the reference plane.
- And let $DE$ be made equal to $AB$.
- And let $BE$, $AE$, and $AD$ have been joined.
- And since $AB$ is at right angles to the reference plane, it will [thus] also make right angles with all straight lines joined to it which are in the reference plane [Def. 11.3] .
- And $BD$ and $BE$, which are in the reference plane, are each joined to $AB$.
- Thus, each of the angles $ABD$ and $ABE$ are right angles.
- So, for the same (reasons), each of the angles $CDB$ and $CDE$ are also right angles.
- And since $AB$ is equal to $DE$, and $BD$ (is) common, the two (straight lines) $AB$ and $BD$ are equal to the two (straight lines) $ED$ and $DB$ (respectively).
- And they contain right angles.
- Thus, the base $AD$ is equal to the base $BE$ [Prop. 1.4].
- And since $AB$ is equal to $DE$, and $AD$ (is) also (equal) to $BE$, the two (straight lines) $AB$ and $BE$ are thus equal to the two (straight lines) $ED$ and $DA$ (respectively).
- And their base $AE$ (is) common.
- Thus, angle $ABE$ is equal to angle $EDA$ [Prop. 1.8].
- And $ABE$ (is) a right angle.
- Thus, $EDA$ (is) also a right angle.
- $ED$ is thus at right angles to $DA$.
- And it is also at right angles to each of $BD$ and $DC$.
- Thus, $ED$ is standing at right angles to the three straight lines $BD$, $DA$, and $DC$ at the (common) point of section.
- Thus, the three straight lines $BD$, $DA$, and $DC$ are in one plane [Prop. 11.5].
- And in which(ever) plane $DB$ and $DA$ (are found), in that (plane) $AB$ (will) also (be found).
- For every triangle is in one plane [Prop. 11.2].
- And each of the angles $ABD$ and $BDC$ is a right angle.
- Thus, $AB$ is parallel to $CD$ [Prop. 1.28].
- Thus, if two straight lines are at right angles to the same plane then the straight lines will be parallel.
- (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
