PROPOSITION IV.

If two straight lines are parallel, the straight line drawn from any point in the one to any point in the other is in the same plane with the parallels.

Let AB, CF be parallel straight lines, P, Q any points. in AB, CF respectively.

Then shall the straight line joining P, Q be in the same plane as AB, CF

For if not, if possible, let it be out of this plane as PXQ, and in this plane draw a straight line PNQ joining P and Q, then there are two straight lines PNQ, PXQ inclosing a space; which is impossible.

Hence, If two straight lines, &c.

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)

DEFINITION.

A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line in that plane which meets it.

PROPOSITION VI..

If a straight line be perpendicular to each of two straight lines at their point of intersection, it shall be perpendicular to the plane passing through them.

Let PQ be to QA and QC. Then shall PQ be to the plane passing through Q4, QC.

Through Qin the same plane as Q4, QC draw any other straight line QH; and through any point H in QH draw a straight line cutting QA, QC in A and C; produce PQ making QR = PQ; join PA, PC, PH, RA, RC, RH.

Then PQ, QA and right▲ PQA

are respectively = RQ, QA and ▲ RQA;

= RA.

R

.. PA

(I. I)

and AC is common to as PCA, RCA ;

.. LPCA = LRCA;

(1. 5)

also PC, CH are respectively = RC, CH;
.. PH=RH;

(I. I)

also PQ, QH are respectively = RQ, QH;

.. ¿PQH = L RQH;

(1.5)

.. PQ is to QH;

.. PQ is to the plane through QA, QC. (Def.)

Conversely :

If a straight line is perpendicular to each of three straight lines which meet in a point, these three straight lines are in one and the same plane.

H

Let PQ be 1 to QA, QC, QX.

Then shall these three straight lines be in one and the same plane.

For if not, if possible, let QX be out of the plane in which are QA, QC, and let the plane in which are PQ, QX cut the former in QH.

Then PQ is to QH,
but it is also to QX;

(by the Prop.)

(hyp.)

which is impossible.

.. If a straight line, &c.

PROPOSITION VII.

If two straight lines be at right angles to the same plane, they shall be parallel to one another.

B

and

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

Join BF and draw FG in the plane BFG1to BF

Then ·.· AB is 1 to the plane BFG, .. LABF is a right 4;

=

.. AB, BF and ABF are respectively GF, FB, and GFB;

Hence AF, FG, GA are respectively = GB, BA, AG ;

.. LAFG = L GBA.

(1.5)

But GBA is a right 4,.. AB is 1 to plane BFG;

also

.. LAFG is a right 4;

CFG is a right 4,. CF is 1 to plane BFG; .. GF is to each of the three straight lines FB, FA, FC; .. FC is in the same plane as AF, FB, in which plane is also AB.

Then.. ABF, CFB are rights;

.. AB is to CF.

(Prop. 6)

(1. 20)