Proof
Suppose the two perpendiculars , met or were skew. Drop
the segment in the plane; the angles at and are right.
In the plane through and , by I.28 the line would
need to be parallel to , fixing the plane through , .
Then within that plane the two right angles force .
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Depends on (4)
- I.28Proposition I.28If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on…
- XI.2Proposition XI.2If two straight lines cut one another, they are in one plane, and every triangle is in one plane.
- XI.3Proposition XI.3If two planes cut one another, their common section is a straight line.
- XI.4Proposition XI.4If a straight line be set up at right angles to two straight lines which cut one another, at their common point of…
Required by (dependents) (3)
- XI.8Proposition XI.8If two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at…
- XI.9Proposition XI.9Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to…
- XI.13Proposition XI.13From the same point two straight lines cannot be set up at right angles to the same plane on the same side.
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