2. In the plane (Prop. 12).
The second case is easily reduced to the first—viz. if by aid of the first we have drawn any perpendicular to the plane from some point without it, we need only draw through the given point in the plane a line parallel to it, in order to have the required perpendicular given. The solution of the first part is of interest in itself. It depends upon a construction which may be expressed as a theorem.
If from a point A without a plane a perpendicular AB be drawn to the plane, and if from the foot B of this perpendicular another perpendicular BC be drawn to any straight line in the plane, then the straight line joining A to the foot C of this second perpendicular will also be perpendicular to the line in the plane.
The theory of perpendiculars to a plane is concluded by the theorem—
Prop. 13. Through any point in space, whether in or without a plane, only one straight line can be drawn perpendicular to the plane.
§ 77. The next four propositions treat of parallel planes. It is shown that planes which have a common perpendicular are parallel (Prop. 14); that two planes are parallel if two intersecting straight lines in the one are parallel respectively to two straight lines in the other plane (Prop. 15); that parallel planes are cut by any plane in parallel straight lines (Prop. 16); and lastly, that any two straight lines are cut proportionally by a series of parallel planes (Prop. 17).
This theory is made more complete by adding the following theorems, which are easy deductions from the last: Two parallel planes have common perpendiculars (converse to 14); and Two planes which are parallel to a third plane are parallel to each other.
It will be noted that Prop. 15 at once allows of the solution of the problem: “Through a given point to draw a plane parallel to a given plane.” And it is also easily proved that this problem allows always of one, and only of one, solution.
§ 78. We come now to planes which are perpendicular to one another. Two theorems relate to them.
Prop. 18. If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane.