556. COR. If a series of lines passing through a common point are cut by two parallel planes, their corresponding segments are proportional.

Hint.-Pass a third plane through the common point parallel to one (and hence the other) of the two given planes.

PROPOSITION XI. THEOREM

557. If two angles not in the same plane have their sides respectively parallel and extending from their vertices in the same direction, they are equal.

GIVEN the angles BAC and B'A'C', whose sides, AB, A'B', and AC,
A'C', are respectively parallel and extending in the same direction.
TO PROVE
angle BAC angle B'A'C'.

Take AB A'B' and AC-A'C' and join AA', BB', CC'.
Then AB' and AC' will be parallelograms.
Hence BB' and CC' are equal to and parallel to AA'.

126

$S 117, 114 Hence BB' and CC' are equal to and parallel to each other.

Ax. 1, 549

Therefore BC' is a parallelogram, and BC= B'C'.
The triangles ABC and A'B'C' are therefore equal.
Hence
angle BAC-angle B'A'C'.

126

89

Q. E. D.

558. COR. If two angles not in the same plane have their sides respectively parallel and extending in opposite directions from their vertices, they are equal; if two corresponding sides extend in the same direction, and the other two in opposite directions, the angles are supplementary.

PROPOSITION XII. THEOREM

559. If two planes are perpendicular to the same straight line, they are parallel.

If they should meet, we should have through any point of their intersection two planes, b and c, perpendicular to the same straight line a.

This is impossible.

Therefore b and c are parallel.

8533

Q. E. D.

560. Exercise.-Prove this proposition as a consequence of $33, 551.

Hint.-Pass two planes through a intersecting b and c in straight lines perpendicular to a.

561. If a straight line is perpendicular to one of two parallel planes, it is perpendicular to the other.

GIVEN the parallel planes MN and PQ, and the line AB perpendicular to MN at A.

TO PROVE

Since AB cuts MN, it also cuts PQ in some point B. § 552 I [If two planes are parallel, any line that cuts one cuts the other.] Through B draw in PQ any straight line BC, and through AB and BC pass a plane intersecting MN in AD.

Then BC is parallel to AD.

§ 544

[If two planes are parallel, their intersections with a third plane are parallel.]

But AB is perpendicular to AD.

$530

[A straight line perpendicular to a plane is perpendicular to every straight line drawn in the plane through its foot.]

Therefore AB is also perpendicular to BC.

836

Since AB is perpendicular to any straight line drawn in

PQ through B, it is perpendicular to PQ.

$530

Q. E. D.

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.

$557 8530

Hence DEF is a right angle.

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.