Cor. 1. Conversely, if two straight lines be parallel, and if one of them be perpendicular to any plane, the other will also be perpendicular to the same plane.

Cor. 2. Two straight lines parallel to a third, are parallel to each other.

For, conceive a plane perpendicular to any one of them, then the other two being parallel to the first, will be perpendicular to the same plane; hence, by the Prop. they will be parallel to each other.

The three straight lines are not supposed to be in the same plane, in this case the Proposition has been already demonstrated.

PROP. IX.

If a straight line, without a given plane, be parallel to a straight line in the plane, it will be parallel to the plane.

Let AB, lying without the plane XZ, be parallel

to CD, lying in the plane,

Then AB is parallel to the plane XZ. Through the parallels AB, CD, draw the plane ABCD.

If the line AB can meet the plane XZ, it must meet it in some point of the line CD, which is the common intersection of the two planes.

But AB cannot meet CD, because AD is parallel to CD.

Hence, AB cannot meet the plane XZ, i. e. AB is parallel to the plane XZ.

PROP. X.

The sections made by a plane cutting two parallel planes, are parallel.

Let FE, GH, be the sections made by the plane GF which cuts the parallel planes XZ, WY;

Then, FE will be parallel to GH.

For if the lines FE, GH, which are situated in the same plane, be not parallel, they will meet if produced. Therefore, the planes XZ, WY, in which these lines lie, will meet if produced, and .. cannot be parallel, which is contrary to the hypothesis.

.. FE is parallel to GH.

PROP. XI.

Z F

W

E

G

H

Y

Parallel straight lines included between two parallel planes are equal.

Let the parallels EG, FH, be cut by the parallel planes XZ, WY, in-the points G, H, E, F,

Then, EG FH,

Through the parallels EG, FH, draw the plane
EGHF, intersecting the parallel planes in GH,
FE.

Then, GH is parallel to FE, by last Prop.
And, GE is parallel to HF;

Z F

W

E

H

G

X.

Cor. Two parallel planes are every where equidistant.

PROP. XII.

If two planes be parallel to each other, a straight line which is perpendicular to one of the planes, will be perpendicular to the other also.

Let the two planes XZ, WY, be parallel, and let the straight line AB, be perpendicular to the plane XZ;

Then, AB will be perpendicular to WY.

For, from any point H in the plane WY, draw HG perpendicular to the plane XZ, and draw AG, BH.

Then, since BA, HG, are both perpendicular to XZ; .. the angles A, G, are right angles.

X

W

Y

And, since the planes XZ, WY, are parallel, .. the perpendiculars BA, HG, are equal.

Hence AG is parallel to BH, and AB being perpendicular to AG, is perpendicular to BH also.

In like manner, it may be proved, that AB is perpendicular to all other lines which can be drawn from B in the plane WY.

.. AB is perpendicular to the plane WY.

Cor. Conversely, if two planes be perpendicular to the same straight line, they will be parallel to each other.

PROP. XIIL

If two straight lines which form an angle, be parallel to two other straight lines which form an angle in the same direction, although not in the same plane with the former, the two angles will be equal, and their planes will be parallel. Let the two straight lines AB, BC, in the plane XZ, be parallel to the two DE, EF, in the plane WY; Then, angle ABC = angle DEF.

For, make AB = DE, BC= EF; join A, C; D, F; A, D; B, E; C, F;

Then, the straight lines AD, BE, which join the equal and parallel straight lines AB, DE, are themselves equal and parallel.

lel.

For the same reason, CF, BE, are equal and paral

W

H

X

Y

.: AD, CF, are equal and parallel, and .. AC, DF, are, also, equal and parallel.

Hence, the two triangles ABC, DEF, having all their sides equal, each to each, have their angles also equal.

.. angle ABC = angle DEF.

Again, the plane XZ is parallel to the plane WY.

For, if not, let a plane drawn through A parallel to DEF, meet the straight lines FC, EB, in G and H.

Then,

But,

which is absurd; hence,

DA = EH = FG Prop.

DA EB = FC

... EH = EB, FG FC

Cor. 1. If two parallel planes XZ, WY, are met by two other planes ADEB, CFEB, the angles ABC, DEF, formed by the intersection of the parallel planes, will be equal.

For the section AB is parallel to the section DE, Prop.

So also, the section BC is parallel to the section EF.

.. angle ABC = angle DEF.

Ꮓ

W

Cor. 2. If three straight lines AD, BE, CF, not situated in the same plane, be equal and parallel, the triangles ABC, DEF, formed by joining the extremities of these straight lines, will be equal, and their planes will be parallel.

PROP. XIV.

If two straight lines be cut by parallel planes, they will be cut in the same

ratio.

Let the straight lines AB, CD, be cut by the parallel planes XZ, WY, VS, in the points A, E, B; C, F, D;

Then, AE EB:: CF: FD. Join A, C; B, D; A, D; and let AD meet the plane WY in G; join E, G; G, F;

Then, the intersections EG, BD, of the parallel planes WY, VS, with the plane ED, are parallel.

(Prop. x.)

.. AE: EB :: AG: GD

Again, the intersections AC, GF, of the parallel planes XZ, YW, with the plane CG, are parallel,

If a straight line be at right angles to a plane, every plane which passes

through it will be at right angles to that plane.

Let the straight line PQ be at right angles to

the plane XZ.

Through PQ draw any plane PO, intersecting XZ in the line OQW.

Then, the plane PO is perpendicular to the plane XZ.

Draw RS, in the plane X7, perpendicular to WQO.

W

S

T

P

X

Then, since the straight line PQ is perpendicular to the plane XZ, it is perpendicular to the two straight lines RS, OW, which pass through its foot in that plane.

But the angle PQR, contained between PQ, QR which are perpendiculars to OW, the common intersection of the planes XZ, PO, measures the angle of the two planes (Def. 5); hence, since this angle is a right angle, the two planes are perpendicular to each other.

Cor. If three straight lines, such as PQ, RS, OW, be perpendicular to each other, each will be perpendicular to the plane of the two others, and the three planes will be perpendicular to each other.

PROP. XVI.

If two planes be perpendicular to each other, a straight line drawn in one of the planes perpendicular to their common section, will be perpendicular to the other plane.

Let the plane VO be perpendicular to the plane XZ and let OW be their common section.

In the plane VO draw PQ perpendicular to OW;

Then PQ is perpendicular to the plane XZ. From the point Q, draw QR in the plane XZ, perpendicular to OW

Then, since the two planes are perpendicular,

V

P

T

W

R

the straight lines QR, QO, which intersect at its foot in the plane XZ. .. PQ is perpendicular to the plane XZ.

X

Cor. If the plane VO be perpendicular to the plane XZ, and if from any point in OW, their common intersection, we erect a perpendicular to the plane XZ, that straight line will lie in the plane VO.

For if not, then we may draw from the same point a straight line in the plane VO, perpendicular to OW, and this line, by the Prop. will be perpendicular to the plane XZ.

Thus we should have two straight lines drawn from the same point in the plane XZ, each of them perpendicular to the given plane, which is absurd.

PROP. XVII.

If two planes which cut each other, be each of them perpendicular to a third plane, their common section will be perpendicular to the same plane.

Let the two planes VO, TW, whose common section is PQ, be both perpendicular to the plane X2.

Then, PQ is perpendicular to the plane XZ. For, from the point Q, erect a perpendicular to the plane XZ.

Then, by Cor, to last Prop., this straight line must be situated at once in the planes VO and TW, and is.. their common section.

T

X

2

SOLID ANGLES.

DEFINITION.

A solid angle is the angular space contained between several planes which meet in the same point.

Three planes, at least, are required to form a solid angle.

A solid angle is called a trihedral, tetrahedral, &c. angle, according as it is formed by three, four, plane angles.

....

PROP. I.

If a solid angle be contained by three plane angles, the sum of any two of these angles will be greater than the third.

It is unnecessary to demonstrate this proposition except in the case where the plane angle, which is compared with the two others, is greater than either of them.

Let A be a solid angle, contained by the three plane angles BAC, CAD, DAB, and let BAC be the greatest of these angles;

Then, CAD + DAB → BAC.

In the plane BAC draw the straight line AE, making the angle BAE = angle BAD.

D

Make AE = AD, and through E draw any straight line BEC, cutting AB, AC, in the points

CE

B, C; join D, B; D, C;

B

Then, ·.· AD = AE, and AB is common to the two triangles DAB, BAE, and the angle DAB = angle BAE.

Again, ·· AD = AE, and AC is common to the two triangles DAC, EAC,

but the base DC base EC.

angle DAC

angle EAC

But, angle DAB = angle BAE

.: angle CAD + angle DAB → angle BAE + angle EAC

> angle BAC.

PROP. II.

The sum of the plane angles which form a solid angle, is always less than four

right angles.

Let P be a solid angle contained by any number of plane angles APB, BPC, CPD, DPE, EPA. Let the solid angle P be cut by any plane ABCDE.

Take any point O in this plane; join A, O; B, O; C, O; D, 0; E, 0;

Then, since the sum of all the angles of every triangle is always equal to two right angles, the

E

P

B