Book 11
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Then, since AD, CB are two parallel straight lines, while one of them, BC, is at right angles to the plane of reference, therefore the remaining one, AD, is also at right angles to the plane of reference. [XI. 8]
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Therefore AD has been set up at right angles to the given plane from the point A in it.
PROPOSITION 13.
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From 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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For, if possible, from the same point A let the two straight lines AB, AC be set up at right angles to the plane of reference and on the same side, and let a plane be drawn through BA, AC; it will then make, as section through A in the plane of reference, a straight line. [XI. 3]
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Let it make DAE; therefore the straight lines AB, AC, DAE are in one plane.
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And, since CA is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference. [XI. Def. 3]
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But DAE meets it and is in the plane of reference; therefore the angle CAE is right.
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For the same reason the angle BAE is also right; therefore the angle CAE is equal to the angle BAE.
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And they are in one plane: which is impossible.
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Therefore etc. Q. E. D.
PROPOSITION 14.
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Planes to which the same straight line is at right angles will be parallel.
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For let any straight line AB be at right angles to each of the planes CD, EF; I say that the planes are parallel.
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For, if not, they will meet when produced.
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Let them meet; they will then make, as common section, a straight line. [XI. 3]
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Let them make GH; let a point K be taken at random on GH, and let AK, BK be joined.
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Now, since AB is at right angles to the plane EF, therefore AB is also at right angles to BK which is a straight line in the plane EF produced; [XI. Def. 3] therefore the angle ABK is right.
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For the same reason the angle BAK is also right.
14
Thus, in the triangle ABK, the two angles ABK, BAK are equal to two right angles: which is impossible. [I. 17]
