562. If a plane is perpendicular to one of two parallel lines, it is perpendicular to the other.

GIVEN the parallel lines AB and DE and the plane MN perpendicular to AB at B.

TO PROVE

Since MN cuts AB, it also cuts DE in some point E. § 548 [If two lines are parallel, any plane that cuts one cuts the other.], Through E draw in MN any straight line EF, and through B draw in MN the line BC parallel to EF.

Then angle DEF=angle ABC.

But, since BC lies in MN, ABC is a right angle.
Hence DEF is a right angle.

$ 557 8530

Since any straight line in MN through E is perpendicular to DE, MN is perpendicular to DE.

Q. E. D.

563. COR. I. If two straight lines are perpendicular to

the same plane, they are parallel.

Hint. Suppose AB and DE perpendicular to MN.

Through any point of DE draw a line, as ED', parallel to AB.

Prove that DE and ED' coincide.

564. Exercise.-Prove & 549 by means of §§ 562, 563.

565. COR. II. The perpendicular distance between two parallel planes is everywhere the same.

DIEDRAL ANGLES AND PROJECTIONS

566. Defs. When two planes meet and are terminated at their common intersection, they are said to form a diedral angle.

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

The faces are regarded as indefinite in extent.

We may designate a diedral angle by two points on its edge and one other point in each face, the former two being written between the latter two. Thus, in the figure, the two planes BC and BD meeting in the line AB form the diedral angle CABD; BC and BD are the faces of the diedral angle, and AB is its edge.

If there is only one diedral angle at an edge, it may be designated by two points on its edge; thus the diedral angle CABD may also be called the diedral angle AB.

567. Def.-The plane angle of a diedral angle is the angle formed by two straight lines drawn one in each face of the diedral angle perpendicular to its edge at the same point.

Thus HKL is the plane angle of the diedral angle CABD.

568. Def.-Two diedral angles are vertical if the faces of one are the prolongations of the faces of the other.

569. Def.-Two diedral angles are adjacent when they have a common edge and a common face lying between them; as ABCD and FBCD.

570. Def. If a plane meets another plane so as to form with it two equal adjacent diedral angles, each of these diedral angles is called a right diedral angle, and the first plane is said to be perpendicular to the second.

Thus the plane PQ is perpendicular to the plane MN, if the diedral angles PQSN and PQSM are equal.

571. If two diedral angles are equal, their plane angles are

GIVEN

TO PROVE

the equal diedral angles MRSP and M'R'S'P'.

their plane angles CAB and C'A'B' equal.

A'

Superpose the diedral angle M'R'S'P' upon its equal MRSP, letting A' fall at A.

Then, since A'B' and AB are both perpendicular to the line RS at A in the plane RP, they coincide.

Similarly A'C' and AC coincide.

Therefore the angles CAB and C'A'B' are equal.

$18

815

Q. E. D.

PROPOSITION XVI. THEOREM

572. If the plane angles of two diedral angles are equal, the diedral angles are equal.

GIVEN two diedral angles, MRSP and M'R'S'P', whose plane angles, CAB and C'A'B', are equal.

TO PROVE

the diedral angles equal.

Since RS is perpendicular to the lines AB and AC, it is perpendicular to their plane.

8531

Similarly R'S' is perpendicular to the plane of A'B' and

A'C'.

Place the angle C'A'B' upon its equal CAB.
Then R'S' will coincide with RS.

$535