Book 11
15
But the angle BGH is right; therefore the angle GBA is also right; therefore GB is at right angles to BA.
15
For the same reason GB is also at right angles to BC.
15
Since then the straight line GB is set up at right angles to the two straight lines BA, BC which cut one another, therefore GB is also at right angles to the plane through BA, BC. [XI. 4]
15
But planes to which the same straight line is at right angles are parallel; [XI. 14] therefore the plane through AB, BC is parallel to the plane through DE, EF.
15
Therefore, if two straight lines meeting one another be parallel to two straight lines meeting one another, not in the same plane, the planes through them are parallel. Q. E. D.
PROPOSITION 16.
16
If two parallel planes be cut by any plane, their common sections are parallel.
16
For let the two parallel planes AB, CD be cut by the plane EFGH, and let EF, GH be their common sections; I say that EF is parallel to GH.
16
For, if not, EF, GH will, when produced, meet either in the direction of F, H or of E, G.
16
Let them be produced, as in the direction of F, H, and let them, first, meet at K.
16
Now, since EFK is in the plane AB, therefore all the points on EFK are also in the plane AB. [XI. 1]
16
But K is one of the points on the straight line EFK; therefore K is in the plane AB.
16
For the same reason K is also in the plane CD; therefore the planes AB, CD will meet when produced.
