Of Planes 
and solid 
A 
Fig. 143. 
Fig, 144, 
Fig. 144, 
Fig. 145, 
226 
7.) Now AB is icular to AD one of two pa- 
nie lines AD, Be, therefore it is perpendicular to Bo b 
the other line, (21.1. Part I.); and since BC is an 
line drawn from B in the plane PQ, it follows that A 
is perpendicular to the plane PQ, (Det. 1.) 
Prop. IX. Teor. 
Parallel straight lines EG, FH intereepted between 
two parallel planes MN, PQ are equal, 
Let a plane EGHF .through the parallel lines, 
so as to meet the el planes in the lines EF, GH; 
these are parallel to each other (7.),.as well as EG, FH; 
therefore EFHG is a parallelogram, and hence EG = 
FH. : 
Cor. Parallel planes are every where at the same 
distance from each other; for if EG and FH are per- 
pendicular to the two planes, they are parallel (1 Cor, 5.), 
and therefore are equal. 
Prop. X. Tueror. 
If two straight lines CA, EA meeting one another, 
be parallel to two others DB, FB that meet one ano- 
ther, though not in the same plane with the first two ; 
the first two and the other two shall contain equal angles; 
and the plane passing through the first two, shall be 
parallel to the plane passing through the other two. 
Take AC = BD, AE = BF, and join CE, DF, AB, 
CD, EF. | Since AC is equal and parallel to BD, the 
figure ABDC is a parallelogram, (28. 1. Part I.) ; 
therefore CD is equal and parallel to AB. For a like 
reason, EF is equal and parallel to AB ; therefore also 
CD is equal and parallel to EF. The figure CEFD is 
therefore a parallelogram, and. thus the side CE is equal 
and parallel to DF; therefore the triangles CAE, DBF 
are equal (]1. 1, Part I.) and the angle CAE=DBF. 
In the next place, the plane ACE is parallel to the 
plane BDF ; for if the plane passing through A paral- 
lel to BDF could meet the two lines DC, FE in any 
other points than C and E, for example in G and H, 
then the three lines AB, DG, FH would be equal (9,), 
and thus DG would be equal to DC, and Fu to FE, 
whieh is absurd ; therefore the plane AEC is parallel 
to BFD. 
Pnop. XI. Tueor. 
If three straight lines AB, CD, EF not situated in 
the same ae are equal and parallel, the triangles 
ACE, BDF formed _b; joining the extremities of these 
lines are equal, and their planes parallel. 
For since AB is equal and parallel to CD, the figure 
ABDC is a parallelogram, therefore the side AC is 
equal and parallel to BD ; in like manner, it may be 
shewn that the sides AE, BF are equal and parallel, as 
also CE, DF; therefore the two triangles CAE, BDF 
are equal ; it may-be demonstrated, as in the last pro- 
position, that their planes are parallel. 
Prop. XII, 
If two straight linés be cut 
shall be cut in the same ratio. 
Teor. 
by parallel planes; they 
Let the straight line AB meet the parallel planes 
GEOMETRY. 
EG, GF, CE geignags mel of invent , 
~ |el } wi ci are paral : 
nan AE:EB::AG:GD, dest art I.) ad F 
«CF: ED. 
MN, PQ, RS in A, E, B; and let the line CD meet Of Ph 
the same planes in C, F, and D; then shall AE: BB: as 
CF: FD ; An 
Draw AD to meet the plane PQ in G, and join AC, 
like masher, We Rea wt >, GE 
therefore AG: GD:: CF: hee ’ 
common ratio AG: GD, we have. "ED 
Prop, XIII. | Tuzor. 
Tf a straight line AP) be perpendicular to a plane Fig. 
MN, fest cr APB, which fasts along’ A, "sal 
be perpendicular to the plane MN, 
Let BC be the intersection of the planes AB, MN. 
In the plane MN draw DE icular to BP; then, 
because AP is dicular to every line drawn from 
P in the plane MN, the angles APD and APB areright 
angles ; but the angle APD formed by the two Pee 
diculars PA, PD is the angle of the planes AB, MN 
(Def. 4.), therefore the two planes are perpendicular to 
one another, (Def. 5. tee ws 
Scuorium. When three straight lines, such as PA, 
PB, PD are perpendicular to each other, each line is 
perpendicular to the plane of the other two, and the 
three planes are perpendicular to one another. .~ 
. « 
CVs 
Prop. XIV. Tuzor. pas 
_ If a plane AB be licular to a plane MN, and Fig. 
in the plane AB a straight line PA be drawn perpend- 
cular to their common intersection PB, the line PA 
shall be perpendicular to the plane MN. ee 
In the plane MN, draw PD perpendicular to PB; 
then because the planes are perpendicular to each other, 
the angle APD is a right angle; therefore AP is per- 
pendicular to the lines PB, PD; consequently it is per- 
pendicular to their plane. 
Cor. If the plane AB be dicular to the plane 
MN, and through P, any point in their common inter- 
section, a perpendicular be drawn to the ple aN, 
this perpendicular shall be in the plane AB. For if it | 
is not, a line AP might be drawn in the plane AB 
perpendicular to PB, the common intersection of 
planes, which at the same time would he pepend cular 
to the plane MN; thus, from the same point P, there 
would be two perpendiculars to a plane MN, which is 
impossible, (2 Cor, 4.) Fhe 
Paor. XV: anon. , 
If two planes AB, AD. be perpendicular to a third pig 
MN;; their common intersection AP is perpendicular to 
the third plane. bs es 
For a perpendicular to the plane MN at P, the a 
in which it meets the two planes AB, AD ae e 
both these at the same time, therefore it is their com- 
mon intersection AP. 
Prop, XVI. -Tuzor. 
If asolid angle be formed by three plane angles, the 
sum of any two of them is greater cages 
