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Book XI.
PROP. XV. THEOR. .
F
F two straight lines meeting one another, be parallel to see N.

two straight lines which meet one another, but are
not in the same plane with the first two; the plane which
passes through these is parallel to the plane passing through
the others.

:

Let AB, BC two straight lines meeting one another, be parallel to DE, EF that meet one another, but are not in the same plane with AB, BC. the planes through AB, BC, and DE, EF shall not meet tho' produced.

From the point B draw BG perpendicular to the plane which a. 11. II. passes through DE, EF, and let it meet that plane in G; and through G draw GH parallel b to ED, and GK parallel to EF. and b. 31.1. because BG is perpendicular to the plane through DE, EF, it shall make right angles with every

E straight line meeting it in that

F plane but the straight lines B

C. 3. Def.ss.
GH, GK in that plane meet it.

K K
therefore each of the angles
BGH, BGK is a right angle.
and because BA is parallel to A

D

d. 9.11. GH (for each of them is paral

H н lel to DE, and they are not both in the same plane with it) the angles GBA, BGH are together equal to two right angles. and BGH is a right angle, therefore c. 29. 1. also GBA is a right angle, and GB perpendicular to BA. for the same reason, GB is perpendicular to BC. since therefore the straight line GB stands at right angles to the two straight lines BA, BC, that cut one another in B; GB is perpendicular f to the plane through f. 4. 11: BA, BC. and it is perpendicular to the plane through DE, EF ; therefore BG is perpendicular to each of the planes through AB, BC and DE, EF. but planes to which the same straight line is perpendicular, are parallel 8 to one another. therefore the plane through g. 14. II. AB, BC is parallel to the plane through DE, EF. Wherefore if two straight lines, &c. Q. E. D.

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Book XI.

PROP. XVI.

THEOR.

See N.

IF

two parallel planes be cut by another plane, their common sections with it

are parallels.

Let the parallel planes AB, CD be cut by the plane EFHG, and
let their common sections with it be EF, GH. EF is parallel to
GH.

For, if it is not, EF, GH Mall meet, if produced, either on the
side of FH, or EG. first, let them be produced on the side of FH,
and meet in the point K. therefore fince EFK is in the plane AB,
every point in EFK is in that
plane; and K is a point in EFK;

K
therefore K is in the plane AB.
for the same reason K is also in
the plane CD. wherefore the
planes AB, CD produced meet

B

D
one another ; but they do not
meet, since they are parallel by
the Hypothesis. therefore the A
straight lines EF, GH do not

E
meet when produced on the
side of FH. in the same manner it may be proved that EF, GH
do not meet when produced on the side of EG. but straight
lines which are in the same plane and do not meet, though pro-
duced either way, are parallel. therefore EF is parallel to GH,
Wherefore if two parallel planes, &c. Q. E. D.

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IF

two straight lines be cut by parallel planes, they shall
be cut in the same ratio. .

Let the straight lines AB, CD be cut by the parallel planes GH,
KL, MN, in the points A, E, B; C, F, D. as AE is to EB, fo is
CF to FD.

Join AC, BD, AD, and let AD meet the plane KL in the point

X; and join EX, XF. because the two parallel planes KL, MN are 1.16.11. cut by the plane EBDX, the common sections EX, BD are parallel'.

for

for the fame reason, because the two parallel planes GH, KL are Book XI. cut by the plane AXFC, the common sections AC, XF are

H

C parallel, and because EX is pa A

G rallel to BD, a side of the triangle ABD, as AE to EB, so is b AX to XD. again, because

b. 2.6. XF is parallel to AC, a side of

L the triangle ADC, as AX to E

F

K
XD, so is CF to FD. and it
was proved that AX is to XD,
as AE to EB, therefore as AE

N
B

D

C. 11. S. to EB, so is CF to FD. Where M fore if two straight lines, &c. Q. E. D.

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F a straight line be at right angles to a plane, every

plane which passes through it shall be at right angles to that plane.

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Let the straight line AB be at right angles to the plane CK. every plane which passes through AB shall be at right angles to the plane CK.

Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK; take any point Fin CE, from which draw FG in the plane D E at

G A Н
right angles to CE, and because
AB is perpendicular to the plane
CK, therefore it is also perpen-

K
dicular to every straight line in
that plane meeting ito, and con-

a. 3. Dcf.11. sequently it is perpendicular to CE, wherefore ABF is a right angle ; but GFB is likewise a

С F B E right angle; therefore AB is parallel to FG. and AB is at right b. 28. 1. angles to the plane CK; therefore FG is also at right angles to the fame plane . but one plane is at right angles to another plane when c. 8. 11. the straight lines drawn in one of the planes, at right angles to their

common

Book XI. common section, are also at right angles to the other planed; and many straight line FG in the plane DE, which is at right angles to 2.4.Def.11. CE the common section of the planes, has been proved to be per

pendicular to the other plane CK; therefore the plane DE is at right angles to the plane CK. In like manner, it may be proved that all the planes which pass through AB are at right angles to the plane CK. Therefore if a straight line, &c. Q. E. D.

PROP. XIX. THEOR.

IF

two planes cutting one another be each of them perpendicular to a third plane; their common section shall be perpendicular to the same plane.

Let the two planes AB, BC be each of them perpendicular to a third plane, and let BD be the common section of the first two. BD is perpendicular to the third plane.

If it be not, from the point D draw, in the plane AB, the straight line DE at right angles to AD the common fection of the plane AB with the third plane; and in the plane BC draw DF at right angles to CD the common section of the plane BC with the third plane. and because the plane AB is perpendicular to the third plane, and DE is drawn in the plane AB at right angles to AD their common

fection, DE is perpendicular to the third a. 4. Def.11. plane'. in the same manner, it

may be pro-
ved that DF is perpendicular to the third
plane. wherefore from the point D two

D
straight lines stand at right angles to the
third plane, upon the same side of it, which A

С 12:

is impossible b. therefore from the point D
there cannot be any straight line at right angles to the third plane,
except BD the common section of the planes AB, BC. BD there-
fore is perpendicular to the third plane. Wherefore if two
planes, &c. Q. E. D.

PROP

Book XI.

PROP. XX.

THEOR.

İF a solid angle be contained by three plane angles, any See N.

see two of them are greater than the third.

Let the solid angle at A be contained by the three plane angles BAC, CAD, DAB. any two of them are greater than the third.

If the angles BAC, CAD, DAB be all equal, it is evident that any two of them are greater than the third. but if they are not, let BAC be that angle which is not less than either of the other two, and is greater than one of them DAB; and at the point A in the straight line AB, make in the plane which passes through BA, AC, the angle BAE equal a to the angle DAB; and make AE equal to a. 23. 1. AD, and through E draw BEC cutting

D
AB, AC in the points B, C, and join
DB, DC. and because DA is equal to
AE, and AB is common, the two DA,
AB are equal to the two EA, AB, and A
the angle DAB is equal to the angle
EAB. therefore the base DB is equal

b. 4. 1. B

Е С to the base BE. and becausc BD, DC are greater than CB, and one of them BD has been proved equal to BE a part of CB, therefore the other DC is greater than the remaining part EC. and because DA is equal to AE, and AC common, but the base DC greater than the base EC; therefore the angle DAC is greater than the angle EAC; and, by the construction, the d. 25.1. angle DAB is equal to the angle BAE; wherefore the angles DAB, DAC are together greater than the angle BAC. but BAC is not less than either of the angles DAB, DAC, therefore BAC with either of them is greater than the other. Wherefore if a solid angle, &c. QE. D.

b

C. 20. I.

PROP. XXI. THEOR.

E VERY solid angle is contained by plane angles which

together are less than four right angles,

First, Let the solid angle at A be contained by three plane angles BAC, CAD, DAB. these three together are less than four right angles.

Take

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