Commentary on Proposition XI.16¶
Back to Proposition XI.16.
Simson points out that, in here quoting I. Def. 23, Euclid should have said “But straight lines in one plane which do not meet in either direction are parallel.”
From this proposition is deduced the converse of XI. 14.
If a straight line is perpendicular to one of two parallel planes, it is perpendicular to the other also.
For suppose that MN, PQ are two parallel planes, and that AB is perpendicular to MN. Through AB draw any plane, and let it intersect the planes MN, PQ in AC, BD respectively. Therefore AC, BD are parallel. [XI. 16] But AC is perpendicular to AB ; therefore AB is also perpendicular to BD. That is, AB is perpendicular to any line in PQ passing through B ; therefore AB is perpendicular to PQ.
It follows as a corollary that
Through a given point one plane, and only one, can be drawn parallel to a given plane.
In the above figure let A be the given point and PQ the given plane. Draw AB perpendicular to PQ. Through A draw a plane MN at right angles to AB (see note on XI. 5 above). Then MN is parallel to PQ. [XI. 14] If there could pass through A a second plane parallel to PQ, AB would also be perpendicular to it. That is, AB would be perpendicular to two different planes through A : which is impossible (see the same note).
Also it is readily proved that,
If two planes are parallel to a third plane, they are parallel to one another.