450. Planes that are perpendicular to the same straight line are parallel.

Given: Planes MN and PQ, each perpendicular to AB;
To Prove: Plane MN is parallel to plane PQ.

For if MN could meet PQ, then through any point common to both would be passed two planes perpendicular to the same straight line AB. But this is impossible (436).

Hence MN cannot meet PQ.

Q.E.D.

PROPOSITION XI. THEOREM.

451. The intersections of two parallel planes with a third plane are parallel.

Given: Two parallel planes, MN, PQ, intersecting plane BD in

AB, DC;

To Prove:

AB is parallel to DC.

Since plane MN cannot meet plane PQ,

(Hyp.)

AB cannot meet DC, though in the same plane BD;

452. COR. Parallel lines that are intercepted between parallel planes are equal.

For since the plane of the parallels AD, BC, intersects the parallel planes MN, PQ, in parallel lines AB, DC, the figure AC is a parallelogram (131); whence AD = BC (136).

PROPOSITION XII. THEOREM.

453. A straight line perpendicular to one of two parallel planes is perpendicular to the other also.

Given: Two parallel planes, MN, PQ, and a straight line AB perpendicular to MN;

To Prove:

AB is perpendicular to PQ.

Through AB pass any plane AD, intersecting MN in AC, and PQ in BD.

Since plane MN is to plane PQ,

(Hyp.)

AC is to BD;

(451)

.. AB is

to AC and BD;

(Hyp., 107)

to plane PQ,

Q.E.D.

(427)

(being

.. AB is
to any line through B in PQ.)

454. COR. Through a given point, A, one plane can be passed parallel to a given plane, PQ, and only one.

For from 4, a perpendicular AB can be drawn to PQ (429); and through 4, a plane can be passed to AB, and hence to plane PQ (450). Moreover, since from 4 but one perpendicular can be drawn to PQ (430), there can be but one plane passed through || to PQ.

455. If two angles not in the same plane have their sides respectively parallel and drawn in the same direction, they are equal.

Given Two angles, BAC, B'A'C', lying in planes MN and PQ, respectively, so that BA and B'A', CA and C'A', are respectively parallel and drawn in the same direction;

To Prove: Angle BAC is equal to angle B'A'C'.

In AB, AC, take any points B and C, and lay off A'B'=AB, A'C' AC; join AA', BB', CC'.

Since AB, AC, are resp. || and = to A'B', A'C',

456. COR. 1. If two angles lying in different planes have their sides respectively parallel, their planes are parallel.

For the intersecting lines that determine the one plane, being parallel to the intersecting lines that determine the other, the planes are parallel.

457. COR. 2. If two parallel planes, MN and PQ, are intersected by two other planes, AB', AC', the angles A, A', formed by their intersections, are equal.

458. COR. 3. If three lines, AA', BB', CC', not in one plane, are equal and parallel, the triangles ABC, A'B'C', formed by joining their extremities, are equal, and their planes are parallel.

PROPOSITION XIV. THEOREM.

459. If two straight lines are cut by three parallel planes, the intercepts are proportional.

Given: A line AB meeting parallel planes MN, PQ, RS, in A, E, B, respectively; and a line CD meeting the same planes in C, G, D, respectively;

Draw AD, meeting PQ in F; join AC, EF, FG, and BD. Since planes PQ, RS, are ||, and plane ABD cuts them,

EF is to BD;

.. AE: EB = AF: FD.

(451) (274)

Since planes PQ, MN, are ||, and plane DAC cuts them,

460. COR. If n straight lines are cut by m parallel planes,

the intercepts are proportional.

461. A dihedral angle is the opening between two planes that meet. The line in which the planes meet is called the edge of the angle, and the two planes are called its faces. Thus the faces AC, BD, meeting in the edge AB, contain the dihedral angle DABC.

To designate a dihedral angle, four letters are generally necessary, two at the edge and one on each face, the two at the edge being placed between the other two. If the edge belongs to only

a

B

one angle, the letters at the edge will suffice to designate the angle. Thus the dihedral angle DABC may be referred to as dihedral angle AB, or simply as the dihedral AB.

462. The plane angle of a dihedral angle is the angle contained by the two perpendiculars drawn, one in each face, to any point in the edge. Thus bac is the plane angle of the dihedral DABC. It is evident that the plane angle is the same at whatever point of the edge it is constructed (455).

A dihedral angle may be conceived as generated by a plane BD turning from coincidence with plane AC about the edge AB as axis, till it reaches the position where its plane angle is

bac; which, again, may be conceived as generated by the revolution of the line ab from an initial position, ac.

463. Two dihedral angles are equal when they can be placed so that their faces coincide.

464. A right dihedral angle has its plane angle a right angle, and its faces are said to be perpendicular to each other. In the same way, dihedral angles are acute or obtuse, and pairs of dihedral angles are adjacent, complementary, supplementary, alternate, corresponding, vertical, etc., according as their plane angles are acute, etc.