S

Again, ABDE; BE = AD, .. AB, BE = AD, DE, each to each, and A E is common to the ▲ ABE, ADE. .. ▲ ABE = / ADE; but ABE is a right,.. ADE is a right, and ED is DA: but ED is to BD and DC: .. ED is to each of the three straight lines, DB, DA, DC, in the point in which they meet... they are in the same plane; but A B is in the plane of BD, DA, three straight lines which meet are in the same plane,.. A B, BD, DC, are in one plane; and each of the s A B D, BD C, is a right. AB is parallel to CD.

PROPOSITION VII.

If two straight lines be 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 A B, CD, be parallel straight lines: take any point E in the one, and any point F in the other: the straight line joining E and F shall be in the plane of the parallels.

E

H

G

B

F

D

For if not, let it, if possible, be above the plane, as EGF: and in the plane A B C D of the parallels: draw the straight line EHF from

E to F: and EGF is also a straight line, the two lines EGF, EHF, include a space, which is impossible. Therefore the straight line joining E and F is not above the plane ABCD; similarly it cannot be below it: it is .. in the plane.

PROPOSITION VIII.

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

Let AB, CD, be the two parallels, and AB to a plane; CD is to the same plane.

Let AB, CD, meet the plane in B and D: join BD: .. AB, CD, B D, are in one plane. (Prop. VII.)

In the plane to which AB is, draw DE

join BE, AE, AD: then: A B is

BD and = A B:

to the plane, each of the ZABE, A BD, is a right ; and BD meets the parallels

AB, CD;

ABD +

CDB two rights; but ABD
CDB is a right : . CD is B D.
DE, and BD is common: and

ABD =

is a right, . And AB ED B, .'. A D = E B. Again, AB = DE, BE = A D, and A E is common. .LABELED A: but / A BE is a right /,.. EDA is so also, and E D is DA: but ED is BD, .. to the plane through B D, DA, but D C is in that plane; .. EDC is a right, or CD is DE, but CD is BD, .. to the plane through B D, DE, i. e., to the same plane to which A B is .

PROPOSITION IX.

Two straight lines, which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another.

Let A B, CD, be each parallel to EF, and not in the same plane with it, A B shall be parallel to c D.

H G K and EF is parallel to A B: .. A B is to HG K. larly CD is to HG K, .. A B and CD being both same plane are parallel. (Prop. VI.)

F

Simi

to the

PROPOSITION X.

If two straight lines meeting one another be parallel to two others that meet one another, and are not in the same plane with the former two, the former two and the other two shall contain equal angles.

Let A B, BC, which meet in B, be respectively parallel to DE,
EF, meeting in E, but not in the same

B

AD, CF, BE, AC, DF: then
and parallel to DE, .. AD is
rallel to BE: for the same reason C F is **
and parallel to BE: .. AD and or being
each of them and parallel to BE, ..
they are and parallel to each other:
extremities towards the same

.. AC and DF, which join their

parts, are also = and parallel. But AB, BC = DE, EF, each to each, and AC = DF, .'. ≤ ABC = ▲ DEF.

PROPOSITION XI.

To draw a straight line perpendicular to a plane from a given point above it.

Let a be the given point, вH the given plane; it is required to draw from the point a, a straight line to BH.

AD to BC.

In the plane draw any straight line BC, and from a draw Then if AD be also to BH, AD is the required: but if not, in BH, draw DEBC: and from a draw af DE: AF is to the plane вH.

Through F draw GH parallel to BC.

Then BC is both to ED and DA, it is to the plane through them; and GH is parallel to BC... GH is also to the

plane EDA and

. GH is to AF: and conversely AF is to GH: and AF is to DE... AF is to the plane through ED and GH, i.e., to the plane вн.

FD a

is

to any line BC in that plane, and AD be joined, that AD to BC.

PROPOSITION XII.

From a given point in a plane to draw a perpendicular to the

plane.

D

B

Let A be the given point in the plane: take any point B above the plane, from which draw BC to the plane: and then from A draw AD parallel to CB: AD is the required. For BC being to the plane, and AD parallel to BC, AD is also to the plane.

PROPOSITION XIII.

To find the perpendicular distance of two lines not in the same

plane.

Let AB, CD, be the two lines. Through any point A in AB, draw AE to CD, and let MN be the plane passing through AB and AE.

From any point D in CD draw DF to MN: and from Fin MN, draw FG || to AE: from G draw GH || to FD: GH is the

For if not, let KI be the shortest distance between AB and CD; then draw KO || to GH and FD. .. Ko is to MN; .. KI is > Ko, but KO = GH, .. KI is > GH, .'. GH is the shortest distance between AB and CD.

In the last three problems, lines are assumed to be drawn from points or to other lines. This assumption, both in this and in other places, is necessary in order that the properties of figures may be exhibited; but the student must be reminded that he has as yet no practical method of drawing those lines, and that the effectual solution of these problems is one of the special objects of this work.

PROPOSITION XIV.

From the same point in a given plane there cannot be two straight lines perpendicular to the plane, upon the same side of it; and there can be only one perpendicular to a plane from a point above the plane.

If possible, let AB, AC, be both to a plane, from the same point A, and upon the same side of the plane.