Book 11
8
And, since AB is equal to DE, and BE to AD, the two sides AB, BE are equal to the two sides ED, DA respectively, and AE is their common base; therefore the angle ABE is equal to the angle EDA.
8
But the angle ABE is right; therefore the angle EDA is also right; therefore ED is at right angles to AD.
8
But it is also at right angles to DB; therefore ED is also at right angles to the plane through BD, DA. [XI. 4]
8
Therefore ED will also make right angles with all the straight lines which meet it and are in the plane through BD, DA.
8
But DC is in the plane through BD, DA, inasmuch as AB, BD are in the plane through BD, DA, [XI. 2] and DC is also in the plane in which AB, BD are.
8
Therefore ED is at right angles to DC, so that CD is also at right angles to DE.
8
But CD is also at right angles to BD.
8
Therefore CD is set up at right angles to the two straight lines DE, DB which cut one another, from the point of section at D; so that CD is also at right angles to the plane through DE, DB. [XI. 4]
8
But the plane through DE, DB is the plane of reference; therefore CD is at right angles to the plane of reference.
8
Therefore etc. Q. E. D.
PROPOSITION 9.
9
Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.
9
For let each of the straight lines AB, CD be parallel to EF, not being in the same plane with it; I say that AB is parallel to CD.
9
For let a point G be taken at random on EF, and from it let there be drawn GH, in the plane through EF, AB, at right angles to EF, and GK in the plane through FE, CD again at right angles to EF.
9
Now, since EF is at right angles to each of the straight lines GH, GK, therefore EF is also at right angles to the plane through GH, GK. [XI. 4]
9
And EF is parallel to AB; therefore AB is also at right angles to the plane through HG, GK. [XI. 8]
