Proposition 12. Theorem.

526. If two straight lines are cut by three parallel planes, the corresponding segments are proportional.

Hyp. Let AB, CD be cut by the M

planes MN, PQ, RS, in the pts. A, E, B,

and C, F, D.

To prove

A

Proof. Join AD cutting the plane

PQ in H.

Join EH and FH.

Then, because the || planes PQ, RS are cut by the plane

ABD, in the lines EH, BD,

1. To draw a perpendicular to a given plane from a given point without it.

2. To erect a perpendicular to a given plane from a given point in the plane.

3. Prove that through a given line of a given plane, only one plane perpendicular to the given plane can be passed.

DIEDRAL ANGLES.

DEFINITIONS.

527. When two planes intersect they are said to forn with each other a diedral angle.

The two planes are called the faces, and their line of in tersection, the edge, of the diedral angle.

Thus, the two planes AC, AE are the A faces, and the intersection AB is the edge, of the diedral angle formed by these planes.

K

B

E

H

A diedral angle is read by the two letters on the edge and one in each face, the two on the edge being read between the other two; or, simply by the two letters on the edge.

Thus, the angle in the figure is read either DABF or AB.

528. If a point is taken in the edge of a diedral angle, and two straight lines are drawn through this point, one in each face, and each perpendicular to the edge, the angle between these lines is called the plane angle of the diedral angle. This plane angle is the same at whatever point of the edge it is constructed.

Thus, if at any point G we draw GH and GK in the two faces AC and AE respectively, and both perpendicular to AB, the angle HGK is equal to the angle DAF, since the sides of these angles are respectively parallel. (525)

The plane of the plane angle HGK is perpendicular to the edge AB (500); and conversely, a plane perpendicular to the edge of a diedral angle at any point cuts the faces in lines perpendicular to the edge. (487)

529. A diedral angle may be conceived to be generated by revolving a plane about a line of the plane.

Thus, suppose a plane, at first in coincidence A, with a fixed plane AC, to turn about the edge AB as an axis until it comes into the position AE; then the magnitude of the diedral angle DABF varies continuously with the amount of turning of this plane about AB. The straight line DA, perpendicular to AB, generates the plane angle DAF.

530. Two diedral angles are equal when one of them can be applied to the other so that the edges coincide, and the two plane faces of the one coincide respectively with the two plane faces of the other.

The magnitude of a diedral angle depends only upon the relative position of its faces, and is independent of their extent.

531. When two diedral angles have a common edge, and the intermediate face common to both, they are said to be adjacent.

Thus, the angles CABD, DABE are adja- B cent angles.

Two diedral angles CABD, DABE, are added together by placing them adjacent to each other, giving as the sum the diedral angle CABE.

532. When the adjacent diedral angles which a plane forms with another plane on opposite sides are equal, each of these angles is called a right diedral angle; and the first plane is said to be perpendicular to the other.

Thus, if the adjacent diedral angles ABCM, ABCN are equal, each of these is a right diedral angle, and the planes AC and MN are perpendicular to each other.

Through a given line in a plane only G one plane can be passed perpendicular to the given plane.

M

E

B

H

N

Proposition 13. Theorem.

533. Two diedral angles are equal if their plane angles are equal.

Hyp. Let the plane s CAD, A C'A'D' of the diedral s AB,

A'B' be equal.

To prove

D

diedral AB = diedral ▲ A'B'. B

Β'

Proof. Apply the diedral / AB to the diedral ▲ A'B', so that the plane CAD coincides with the equal plane ZC'A'D'.

Because the pt. A coincides with the pt. A', and the plane CAD with the plane C'A'D',

(485)

.. AB, the to the plane CAD, coincides with A'B', the to the plane C'A'D'.

(501)

Because AB coincides with A'B', and AC with A'C', .. the plane BC coincides with the plane B'C'. (485)

Similarly, the plane BD coincides with the plane B'D'. .. diedral

AB = diedral A'B'.

(530)

Q.E. D.

534. SCH. A diedral angle is called acute, right, or obtuse according as the corresponding plane angle is acute, right, or obtuse.

EXERCISE.

Prove that through a line parallel to a given plane, only one plane perpendicular to the given plane can be passed.

Proposition 14. Theorem.

535. The ratio of any two diedral angles is equal to the ratio of their plane angles.

Hyp. Let CABD, C'A'B'D' be two diedrals, and let CAD, C'A'D' be their planes.

To prove

AB

CAD

A'B'C'A'D"

Proof. Take any common measure of the ≤s CAD, C'A'D', and suppose it to be contained three times in CAD and 4 times in C'A'D'.

Pass planes through the edges AB, A'B' and the several lines of division of the plane s CAD, C'A'D' made by the common measure.

These planes divide CABD into 3, and C'A'B'D' into 4 equal parts.

(533)

... CABD: C’A’B’D′ = 3 : 4.

... CABD : C'A'B'D' = < CAD : C'A'D'. This proof may be extended to the case where the plane angles are incommensurable, by the method employed in (234), (298), and (356).

Q. E.D.

536. SCH. Since the diedral angle and its plane angle

increase and decrease in the same ratio, the plane angle is taken as the measure of the diedral angle.

See (236).