Proposition 10. Theorem.

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

Hyp. Let MN and PQ be || planes, and let AB be to PQ.

M

A

C

And since AC is any line drawn through A in the plane

MN,

.. AB is to the plane MN.

(487)

Q.E. D.

522. COR. 1. Through a given point one plane can be passed parallel to a given plane, and only one.

For, if AB is to PQ, a plane passing through the pt. A, L to AB, is || to PQ. (516)

Also, since every plane | to PQ is to AB (521), and since only one plane can be passed through the pt.A to AB (504), therefore only one plane can be passed through a given pt. parallel to a given plane.

523. COR. 2. Two parallel planes are everywhere equally distant.

For, all st. lines to the plane PQ are also to the li plane MN (521); and being to the same plane, they are || (509); and being included between || planes, they are equal (520). Hence the planes are everywhere equally distant.

(497)

524. COR. 3. If two intersecting straight lines are each parallel to a given plane, the plane of these lines is parallel to the given plane.

See (518), (516).

Proposition 11. Theorem.

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

Hyp. Let s A and A' lie in the planes MN, PQ respectively, and let AB be to A'B' and AC be || to A'C'. ZA = ZA'.

(1) To prove Proof. Take AB = A'B', and AC = A'C', and join AA', BB', CC', BC,

M

A

P

Because BB' and CC' are each = and || to AA',

.. BB' is = and || to CC'.

(511)

(133)

.. AABCAA'B'C',

having the three sides equal each to each (108).

(2) To prove

.. ZA ZA'.

=

MN || PQ.

Proof. Since the lines AB, AC are each || to the plane

Find the locus of points equally distant from three given

points.

Proposition 12. Theorem.

526. If two straight lines are out 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,

Α

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 form with each other a diedral angle.

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

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

G

H

K

B

'C

E

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.

B

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

B

H

D