Conversely:

If two straight lines be parallel, and one of them be at right angles to a plane, the other shall also be at right angles to the same plane.

Let AB be to CF, and CF be 1 to the plane BFG.
Then shall AB be 1 to the plane BFG.

Join BF, and in the plane BFG draw GF1 to BF and = AB. Join BG, AG, AF.

Then . CF is 1 plane BFG, .. LS CFB, CFG are rights;

and AB is I to CF, .. LS ABF, CFB are together two right s;.. ABF is a right 4,

=

(I. 21)

and GF is 1 to CF, and also 1 to BF; .. GF is 1 to the plane in which are CF, BF (Prop. 6), and .. 1 to AF, .. LAFG is a right .

Again, AB, BF and ▲ ABF are respectively = GF, FB and GFB;

.. AF = GB.

(I. I) Hence AF, FG, GA are respectively GB, BA, AG;

=

(1.5)

and also LABF is a right 4;

.. LABG is a right 4,

.. AB is to the plane BFG.

(Prop

PROPOSITION VIII.

It is always possible to draw a straight line perpendicular to a plane from a given point without it.

K

For let AB be a given plane, and P a point without it, and let it be required to draw from P a straight line 1 to the plane AB.

In the plane AB draw any straight line CF, and from Plet fall PG 1 to CF.

Then if PG is also to the plane AB, the thing required has been done.

But, if not, in the plane AB draw GQ1 to CF, and from P let fall PQ1 to GQ; then PQ is 1 to the plane AB.

Through Q draw HK || to CF.

Then. CF is 1 PG and also 1 QG;

... CF is to plane PGQ.

(Prop. 6)

But HK is

and is..

to CF, ... HK is 1 to plane PGQ, (Prop. 7) to PQ;

.. PQ is to HK, and PQ is also to QG;

.. PQ is to plane AB.

(Prop. 6)

PROPOSITION IX.

There cannot be drawn more than one straight line perpendicular to a plane from a given point without it.

For, if possible, let PQ, PR be each of them to the plane AB.

Join QR.

Then PQR, PRQ are each of them right angles;

which is impossible.

(1. 13)

PROPOSITION X.

It is always possible to draw a straight line perpendicular to a plane from a given point within it.

For let AB be a given plane, and C a point within it, and let it be required to draw from C a straight line 1 to the plane AB.

From any point P without the plane let PQ be drawn 1 to the plane.

Through C draw CF || to PQ.

Then shall CF be 1 to the plane AB.

For if two straight lines are I and one of them is to a plane, the other is also to the same plane. (Prop. 7)

PROPOSITION XI.

There cannot be drawn more than one straight line perpendicular to a plane from a given point within it.

A

F

For, if it be possible, from a point C within the plane AB, let two straight lines CF, CG be drawn to the plane AB, and let the plane through CF, CG cut the plane AB in XY.

Then s FCY, GCY are each rights, and are .. equal to one another; which is impossible.