In the plane CD draw any straight line BC cutting AD. This line cuts PQ, and therefore cuts MN, by the first part of the proposition.

Therefore the plane CD, in which BC lies, will cut MN.

Q. E. D.

553. COR. I. If two planes are parallel to a third plane they are parallel to each other.

554. COR. II. Through a given point without a given plane there can be drawn a plane parallel to the given plane, and

Hint.-Through the point A, without the plane MN, draw two straight lines AB and AC parallel to MN.

PQ, the plane of AB and AC, will be parallel to MN.

No other plane through A could be parallel to MN, for it would cut PQ. and therefore also MN.

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

GIVEN the straight lines AB and CT cut by the parallel planes MN, PQ, and RS in the points A, E, B, and C, H, T.

Join A to T by a straight line cutting PQ in G.

Draw EG, BT, GH, and AC.

Then EG and GH will be parallel to BT and AC respec

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'.

§ 126

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'.

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 di rections, the angles are supplementary.

PROPOSITION XII. THEOREM

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

GIVEN the planes b and c perpendicular to the straight line a.
TO PROVE

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. $5521 [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.

$36

Since AB is perpendicular to any straight line drawn in

PQ through B, it is perpendicular to PQ.

$530

Q. E. D