Book XI.

PRO P. XV.

THE OR.

IF
F two straight lines meeting one another, be parallel to See N.

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

E

Let AB, BC, two straight lines meeting one another, be paa rallel to DE, EF that meet one another, but are not in the. fame plane with AB, BC: The planes through AB, BC, and DE, EF shall not meet, though produced.

From the point B draw BG perpendicular to the plane a 11. II. which pafses through DE, EF, and let it meet that plane in G; and through G draw GH parallel b to ED, and GK pa. b 31. I. sallel to EF : And because BG is perpendicular to the plane through DE, EF, it thall make right angles with every

F straight line meeting it in that B

G plane : But the straight lines

¢ 3. def. 11. C

K K CH, GK in that plane meet it: Therefore each of the angles BGH, BGK is a right angle : And becaufe BA is A

D parallel d to GH (for each of

H

de ir. them is parallel to DE, and they are not both in the fame plane with it) the angles GBA, BGH are together equal to two right angles : And BGH is a c 29. 1. right angle; therefore also GBA is a right angle, and GB per pendicular to BA; For the same reason, GB is perpendicular to BC: Since therefore the straight line GB stands at right angles to the two ftraight lines BA, BC, that cut one another in B; GB is perpendicular ' to the plane through BA, BC: And f 4. II. 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 to one another : Therefore the plane through AB, 8 14. 11. BC is parallel to the plane through DE, EF. Wherefore, if two straight lines, &c. Q E. D).

.

Book XI.

PRO P. XVI.

THE OR

Sec 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 shall meet, if produced, either on
the side of FH, or EG : First, let them be produced on the fide
of FH, and meet in the point K: Therefore, Gince EFK is in
the plane AB, every point in
EFK is in that plane; and

K
K is a point in EFK ; there.
fore K is in the plane AB:
For the same reason K is also

F

АН
in the plane CD: Wherefore
the planes AB, CD produced

B

D meet one another ; but they do not meet, since they are parallel by the hypothesis :

A

с
Therefore the straight lines E
EF, GH do not meet when

G produced on the Gde 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 produced either way, are parallel: Therefore EF is parallel to GH. Wherefore, if two parallel planes, &c, Q. E. D.

IF F 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, so 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 cut by the plane EBDX, the common sections

EX, BD, are parallela. For the same reason, because the two Book XI. parallel planes GH, KL are eut by the plane AXFC, the

H a 16.11.

1C common fections. AC, XF are A parallel : And because EX is G parallel to BD, a side of the triangle ABD, as AE to EB, so is b AX to XD. Again, be

L

b 2. 6. cause XF is parallel to AC, a fide of the triangle ADC, as

E

K
AX to XD, so is CF to FD:
And it was proved that AX is
to XD, as AE to EB ; There-

N fore c'; as AE to EB, so is CF

B

D

CII. . to FD. Wherefore, if two

M straight lines, &c. Q. E, D,

IE
E a straight line be at right angles to a plane, every

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

Let the straight line AB be at right angles to a plane CK; every plane which pafses 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 F in CE, from which draw FG in

D G
the plane DE at right angles

A H
to CE; And because AB. is
perpendicular to the plane
CK, therefore it is also per-

K
pendicular to every straight
line in that plane meeting it a;

a 3. dcf, its
And consequently it is perpen.
dicular 10 CE : Wherefore
ABF is a right angle; but C F B E
GFB is likewise a right angle

; therefore AB is parallelo to FG. And AB is at right angles to b 28. I, the plane CK ; therefore FG is also at right angles to the same planec. But one plane is at right angles to another plane when c 8. II. ike straight lines drawn in one of the planes, at right angles

to

O 2

Book XI. to their common fection, are also at right angles to the other

plane d; and any straight line FG in the plane DE, which is at d 4. def. II.

right angles to CE the common seation of the planes, has been proved to be perpendicular 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.

IF two planes cutting one another be each of them per

pendicular to a third plane ; their common fection 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 feètion 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 section 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

B
plane AB is perpendicular to the third
plane, and DL is drawu in the plane AB
at right angles to AD their common

section, DE is perpendicular to the third EF 3 4. def. 11. plane a. In the same manner, it may

be proved that DF is perpendicular to
the third plane. Wherefore, from the
point D two straight lines stand at right

angles to the third plane, upon the fame
b 13. 11. fide of it, which is impoflible b: There- D

fore, from the point D there cannot be
any straight line at right angles to the A

С
third plane, except BD the common fece
tion of the planes AB, BC. BD therefore is perpendicular to
the third plane. Wherefore, if (wo planes, &c. Q. E. D.

P O P

Book XI.

PRO P. XX. THE O R. a solid angle be contained by three plane angles, any See N. 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 to thea 23. 1.
angle DAB; and make AE equal to AD, and through E
draw BEC cutting AB, AC in the

D
points B, C, and join DB, DC. And
because DA is equal to AE, and AB
is common, the two DA, AB are e-
qual to the two EA, AB, and the
angle DAB is equal to the angle EAB:
Therefore the base DB is equalb to

b 4. I. the base BE. And because BD, DC B

Е С 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 d. 25. 1. the angle EAC; and, by the construction, the angle DAB is equal to the angle BAE; wherefore the angles DAB, DAC are together greater than BAE, EAC, that is, 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. Q. E. D.

C 20. Ia

E E

VERY solid angle is contained by plain angles which
together are less than four right angles.

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