Book 11
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For let the two planes AB, BC be at right angles to the plane of reference, and let BD be their common section; I say that BD is at right angles to the plane of reference.
19
For suppose it is not, and from the point D let DE be drawn in the plane AB at right angles to the straight line AD, and DF in the plane BC at right angles to CD.
19
Now, since the plane AB is at right angles to the plane of reference, and DE has been drawn in the plane AB at right angles to AD, their common section, therefore DE is at right angles to the plane of reference. [XI. Def. 4]
19
Similarly we can prove that DF is also at right angles to the plane of reference.
19
Therefore from the same point D two straight lines have been set up at right angles to the plane of reference on the same side: which is impossible. [XI. 13]
19
Therefore no straight line except the common section DB of the planes AB, BC can be set up from the point D at right angles to the plane of reference.
19
Therefore etc. Q. E. D.
PROPOSITION 20.
20
If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.
20
For let the solid angle at A be contained by the three plane angles BAC, CAD, DAB; I say that any two of the angles BAC, CAD, DAB, taken together in any manner, are greater than the remaining one.
20
If now the angles BAC, CAD, DAB are equal to one another, it is manifest that any two are greater than the remaining one.
20
But, if not, let BAC be greater, and on the straight line AB, and at the point A on it, let the angle BAE be constructed, in the plane through BA, AC, equal to the angle DAB; let AE be made equal to AD, and let BEC, drawn across through the point E, cut the straight lines AB, AC at the points B, C; let DB, DC be joined.
20
Now, since DA is equal to AE, and AB is common, two sides are equal to two sides; and the angle DAB is equal to the angle BAE; therefore the base DB is equal to the base BE. [I. 4]
