Unite the three remaining ends in a knot and pull the strings taut.

A line through the given point and the knot is the perpendicular required. Prove the method correct by supposing if possible that the foot of the perpendicular from the knot is not in the given point, and apply § 103.

II. To draw a perpendicular to a given plane from a given point without it.

From the point with a string of convenient length measure three equal distances to the plane.

The centre of the circumference which passes through the three points thus found is the foot of the required perpendicular. (Why?)

PARALLEL LINES AND PLANES

542. Def.-A straight line and a plane are parallel to each other if they cannot meet, however far produced. 543. Def.-Two planes are parallel to each other if they cannot meet, however far produced.

544. If two parallel planes are cut by a third plane, their intersections with this plane are parallel.

GIVEN the parallel planes MN and PQ cut by the plane AD in the lines AC and BD.

Since the planes MN and PQ cannot meet, the lines AC and BD lying in them cannot meet.

Moreover these lines lie in the same plane AD.

Therefore they are parallel.

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Q. E. D.

545. COR. Parallel lines AB and CD intercepted between parallel planes are equal.

PROPOSITION VII. THEOREM

546. If a straight line is parallel to a plane, the intersection of the plane with a plane passed through the line is par

GIVEN the line BA parallel to the plane MN and a plane BF passing through BA and intersecting MN in EF.

These lines lie in the same plane.

They cannot meet, for BA cannot meet the plane MN in

which EF lies.

Therefore they are parallel.

$ 31

Q. E. D

(2)

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.

PROPOSITION VIII. THEOREM

548. A plane which cuts one of two parallel lines must, if sufficiently produced, cut th 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 OX. 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.

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

Therefore MN must cut CD.

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.

Hint.-Pass a plane through a and any point of c.

This plane will entirely contain c. Otherwise it would cut 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.

and there

This con

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 S 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.