PROPOSITION XII. THEOREM

487. If two straight lines are cut by parallel planes, their corresponding segments are proportional.

Hyp. Parallel planes MN, PQ, and RS are intersected by two straight lines in A, B, C and D, E, F, respectively.

Proof. Draw AF, and pass a plane through AC and AF, intersecting the planes PQ and RS in BG and CF, respectively.

Then

CF is to BG.

(478)

By passing a plane through AF and DF, it follows, similarly,

488. COR. If through any point straight lines are drawn

intersected by two parallel planes; their corresponding segments are proportional.

PROPOSITION XIII. THEOREM

489. Planes perpendicular to the same straight line are parallel to each other.

Hyp. Planes MN and PQ are perpendicular to line AB.

To prove

MN is to PQ.

HINT. Apply the indirect method and (98).

PROPOSITION XIV. THEOREM

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

Hyp. Plane MN is || to plane PQ, and AB is to plane MN.

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

Therefore AB is perpendicular to any line in PQ passing through B. Whence

AB is to plane PQ.

Q.E.D.

491. COR. Through a given point, one plane and only one may be passed parallel to a given plane.

Ex. 1046. If two planes are parallel to a third plane, they are parallel to each other.

DIEDRAL ANGLES

492. DEF. A diedral angle is the opening between two planes intersecting in a straight line.

The edge is the line of intersection, and the faces are the intersecting planes.

Thus, in the diedral angle formed by the planes AC and BE, BC is the edge and AC and BE are the faces.

493. A diedral angle may be designated by two letters on its edge; or if several diedral angles have a common edge, by four letters, one on each face, and two on the edge, the letters on the edge being placed between the others.

Thus, the diedral angle in the annexed diagram may be designated by BC or ABCE.

B

494. DEF. The plane angle of a diedral angle is the angle formed by perpendiculars to the edge at some point, one in each face.

Thus, CDE is a plane angle of diedral angle AB if CD is perpendicular to AB and ED is perpendicular to AB.

Two plane angles of the same diedral angle, as CDE and FGH, are equal, since their sides are respectively parallel.

Or, the plane angle of a diedral angle is the same, at whatever point of the edge it is drawn.

A

G

H

B

The size of a diedral angle does not depend upon the extent of its faces, but upon the difference of their positions.

495. Two diedral angles are equal when they can be made to coincide.

By superposing two equal diedral angles, their plane angles can be made to coincide.

496. A diedral angle is acute, right, obtuse, or straight, according as its plane angle is acute, right, obtuse, or straight. It is evident that the two faces of a straight diedral angle lie in the same plane.

Diedral angles are complementary, supplementary, adjacent, etc., according as their plane angles are complementary, supplementary, adjacent, etc.

If a plane meets another plane so as to make the two adjacent diedrals equal, each diedral is evidently a right one.

497. Planes forming right diedral angles are said to be perpendicular to each other. Thus, plane CD is

perpendicular to plane AB if they form right diedral angles.

498. DEF. The projection of a point on a plane is the foot of the perpendicular drawn from that point upon the plane. The projection of a line upon a plane is the locus of the projection of all the points of the line.

PROPOSITION XV. THEOREM

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

B

C

Hyp. The diedral angles AB and A'B' have equal plane angles CBD and C'B'D' respectively.

To prove diedral angle AB

=

diedral angle A'B'.

Proof. Since AB is 1 to BD and 1 to BC,

AB is to plane BDC.