547. COR. If two intersecting straight lines are parallel to a plane, their plane is parallel to the given plane.

Hint.-If their plane were not parallel to the given plane it would intersect it in a line which would be parallel to both the given lines.

548. A plane which cuts one of two parallel lines must, if sufficiently produced, cut the other also.

GIVEN the parallel lines AB and CD, one of which, AB, is cut by the plane MN in the point O.

TO PROVE

Pass a plane through AB and CD.

As this plane and the plane MN have the point in common, their intersection must contain O. Call it O.X. Now suppose, if possible, that MN does not cut the line CD, but is parallel to it.

Then OX will also be parallel to CD.

And there will be two lines, OX and OB through O, allel to CD, which is impossible.

Therefore MN must cut CD.

$546

par

Q. E. D.

549. COR. I. If two straight lines a and c are parallel to a third b, they are parallel to each other.

This plane will entirely contain c.

Otherwise it would cut

and there

This con

fore b, which is parallel to c, and also a, which is parallel to b.

tradicts the hypothesis that it contains a.

Prove also that a and c cannot meet.

550. COR. II. If two straight lines a and b are parallel, any plane MN, that contains one, as b, and not the other, is parallel to the second.

Hint.-If MN is not parallel to a, it will cut it.
This is impossible, for then MN would cut b also.
Therefore MN is parallel to a.

551. COR. III. If two intersecting straight lines are parallel to two other intersecting straight lines, the plane of the first pair is parallel to the plane of the second pair.

Hint.-Apply § 550 and then § 547.

PROPOSITION IX. THEOREM

552. If two planes are parallel:

I. Any straight line that cuts one cuts the other.
II. Any plane that cuts one cuts the other.

I. GIVEN the parallel planes MN and PQ and the straight line AF cutting PQ in the point A.

TO PROVE that AF is not parallel to MN but cuts MN.

Through AF and any point A' of MN not in AF pass a plane A'B.

Since this plane has a point in common with each of the parallel planes, it will intersect each in straight lines AB and A'B'. These lines will be parallel.

$ 544

In the plane A'B we have AF cutting AB, one of the two parallels AB and A'B'.

It therefore cuts the other, A'B', since AF and AB cannot both be parallel to A'B'.

Ax. b

Therefore AF cutting A'B' cuts the plane MN in which A'B' lies.

Q. E. D.

TO PROVE

that CD also intersects MN.

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 MV.

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.