Kitap 11
9
For the same reason CD is also at right angles to the plane through HG, GK; therefore each of the straight lines AB, CD is at right angles to the plane through HG, GK.
9
But if two straight lines be at right angles to the same plane, the straight lines are parallel; [XI. 6] therefore AB is parallel to CD.
PROPOSITION 10.
10
If two straight lines meeting one another be parallel to two straight lines meeting one another not in the same plane, they will contain equal angles.
10
For let the two straight lines AB, BC meeting one another be parallel to the two straight lines DE, EF meeting one another, not in the same plane; I say that the angle ABC is equal to the angle DEF.
10
For let BA, BC, ED, EF be cut off equal to one another, and let AD, CF, BE, AC, DF be joined.
10
Now, since BA is equal and parallel to ED, therefore AD is also equal and parallel to BE. [I. 33]
10
For the same reason CF is also equal and parallel to BE.
10
Therefore each of the straight lines AD, CF is equal and parallel to BE.
10
But straight lines which are parallel to the same straight line and are not in the same plane with it are parallel to one another; [XI. 9] therefore AD is parallel and equal to CF.
10
And AC, DF join them; therefore AC is also equal and parallel to DF. [I. 33]
10
Now, since the two sides AB, BC are equal to the two sides DE, EF, and the base AC is equal to the base DF, therefore the angle ABC is equal to the angle DEF. [I. 8]
10
Therefore etc.
PROPOSITION 11.
11
From a given elevated point to draw a straight line perpendicular to a given plane.
11
Let A be the given elevated point, and the plane of reference the given plane; thus it is required to draw from the point A a straight line perpendicular to the plane of reference.
11
Let any straight line BC be drawn, at random, in the plane of reference, and let AD be drawn from the point A perpendicular to BC. [I. 12]
11
If then AD is also perpendicular to the plane of reference, that which was enjoined will have been done.
