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equal to EH, and AG to BH: and because AE is equal to EB, Book XI. and FE common and at right angles to them, the base AF is an equal b to the base FB; for the same reason, CF is equal to FD : b 4. 1. and because AD is equal to BC, and AF to FB, the two sides FA, AD are equal to the two FB, BC, each to each ; and the base DF was

F proved equal to the base FC; therefore the angle FAD is equal d to the angle

d 8.1. FBC : again, it was proved that GA is equal to BH, and also AF to FB; FA,

C then, and AG are equal to FB and BH, A and the angle FAG has been proved equal to the angle FBH; therefore the G base GF is equal to the base FH: again, because it was proved, that GE is equal

E

H to EH, and EF is common ; CE, EF are equal to HE, EF ; and the base GF is

D

B equal to the base FH; therefore the angle GEF is equal d to the angle HEF; and consequently each of these angles is a right e angle. Therefore FE makes right an-e 10. def. gles with GH, that is, with any straight line drawn through E in 1 the plane passing through AB, CD. In like manner, it may be proved, that FE makes right angles with every straight line which meets it in that plane. But a straight line is at right angles to a plane when it makes right angles with every straight line which meets it in that plane f : therefore EF is at right an- f 3. def. gles to the plane in which are AB, CD. Wherefore, if a straight 11. line, &c. Q. E: D.

PROP. V. THEOR.

IF three straight lines meet all in one point, and a See N. straight line stands at right angles to each of them in that point; these three straight lines are in one and the same plane.

Let the straight line AB stand at right angles to each of the straight lines BC, BD, BE, in B the point where they meet ; BC, BD, BE are in one and the same plane.

If not, let, if it be possible, BD and BE be in one plane, and BC be above it; and let a plane pass through AB, BC, the common section of which with the plane, in which BD and BE are,

Book XI. shall be a straight a line ; let this be Br: therefore the three

straight lines AB, BC, BF are all in one plane, viz. that which a 3. 11. passes through AB, BC; and because AB stands at right angles

to each of the straight lines BD, BE, it is also at right angles b 4. 11. b to the plane passing through them ; and therefore makes right c 3. def. angles c with every straight line meet11. ing it in that plane ; but BF which is А,

in that plane meets it: therefore the
angle ABF is a right angle ; but the

C
angle ABC, by the hypothesis, is also

F
a right angle; therefore the angle ABF
is equal to the angle ABC, and they
are both in the same plane, which is

D impossible: therefore the straight line

B
BC is not above the plane in which are

E
BD and BE: wherefore the three
straight lines BC, BD, BE are in one and the same plane. There-
fore, if three straight lines, &c. Q. E. D.

PROP. VI. THEOR.

IF two straight lines be at right angles to the same plane, they shall be parallel to one another.

Let the straight lines AB, CD be at right angles to the same plane ; AB is parallel to CD.

Let them meet the plane in the points B, D, and draw the straight line BD, to which draw DE at right angles, in the same plane ; and make DE equal to AB, and join

A

С BE, AE, AD. Then, because AB is pera 3. def. pendicular to the plane, it shall make right a 11.

angles with every straight line which meets
it, and is in that plane : but BD, BE, which
are in that plane, do each of them meet AB.
Therefore each of the angles ABD, ABE is

B В

D a right angle: for the same reason, each of the angles CDB, CDE is a right angle : and because AB is equal to DE, and BD common, the two sides AB, BD are equal to the two ED, DB; and they contain right

E b 4. 1. angles; therefore the base AD is equal to the base BE : again,

because AB is equal to DE, and BE to AD; AB, BE are equal

to ED, DA; and, in the triangles ABE, EDA, the base AE is Book XI. common; therefore the angle ABE is equal to the angle EDA: but ABE is a right angle; therefore EDA is also a right angle, c 8. 1. and ED perpendicular to DA: but it is also perpendicular to each of the two BD, DC: wherefore ED is at right angles to each of the three straight lines BD, DA, DC in the point in which they meet: therefore these three straight lines are all in the same plane d: but AB is in the plane in which are BD, DA, because d 5. 11.. any three straight lines which meet one another are in one planee; e 2. 11. therefore AB, BD, DC are in one plane: and each of the angles ABD, BDC is a right angle; therefore AB is parallelf to ČD. f 28. 1. Wherefore, if two straight lines, &c. Q. E. D.

PROP. VII. THEOR. i.

IF two straight lines be parallel, the straight line See N. drawn from any point in the one to any point in the other is in the same plane with the parallels.

Let AB, CD be parallel straight lines, and take any point E in the one, and the point F in the other: the straight line which joins E and F is in the same plane with the parallels.

If not, let it be, if possible, above the plane, as EGF; and in the plane ABCD in which the paral

A
E

B
lels are, draw the straight line EHF
from E to F; and since EGF also
is a straight line, the two straight

G lines EHF, EGF include a space be

H tween them, which is impossible a. Therefore the straight line joining

Ax, 1. the points E, F is not above the C

F D plane in which the parallels AB, CD are, and is therefore in that plane. Wherefore, if two straight lines, &c. Q. E. D.

a 10,

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IF two straight lines be parallel, and one of them See N. is at right angles to a plane, the other also shall be at right angles to the same plane..

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Book XL. Let AB, CD be two parallel straight lines, and let one of them

AB be at right angles to a plane; the other CD is at right angles
to the same plane.

Let AB, CD meet the plane in the points B, D, and join BD: 87.11. therefore & AB, CD, BD are in one plane. In the plane to which

AB is at right angles, draw DE at right angles to BD, and make
DE equal to AB, and join BE, AE, AD. And because AB is per-

pendicular to the plane, it is perpendicular to every straight line a 3.

which meets it, and is in that plane a : therefore each of the angles def. 11. ABD, ABE is a right angle: and because the straight line BD

meets the parallel straight lines AB, CD, the angles ABD, CDB b 29. 1. are together equal b to two right angles: and ABD is a right an

gle; therefore also CDB is a right angle, and CD perpendicular
to BD: and because AB is equal to DE, and BD common, the
two AB, BD, are equal to the two ED,

с
DB, and the angle ABD is equal to the

А angle EDB, because each of them is a

right angle; therefore the base AD is C 41.

equal c to the base BE: again, because AB
is equal to DE, and BE to AD; the two
AB, BE are equal to the two ED, DA;
and the base AE is common to the trian-

B

D
gles ABE, EDA; wherefore the angle
d 8. 1.

ABE is equald to the angle EDA: and
ABE is a right angle; and therefore EDA
is a right angle, and ED perpendicular to

DA: but it is also perpendicular to BD; E
e 4. 11. therefore ED is perpendiculare to the
f 3. def. plane which passes through BD, DA, and shall make right an-
11.

gles with every straight line meeting it in that plane: but DC is
in the plane passing through BD, DA, because all three are in
the plane in which are the parallels AB, CD: wherefore ED is
at right angles to DC; and therefore CD is at right angles to
DE: but CD is also at right angles to DB; CD then is at right
angles to the two straight lines DE, DB in the point of their in-
tersection D; and therefore is at right angles to the plane pass-
ing through DE, DB, which is the same plane to which AB is at
right angles. Therefore, if two straight lines, &c. Q. E. D.

Book XI.

PROP. IX. THEOR.

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 AB, CD be each of them parallel to EF, and not in the same plane with it; AB shall be parallel to CD.

In EF take any point G, from which draw, in the plane passing through EF, AB, the straight line GH at right angles to EF; and in the plane passing through EF, CD, draw GK at right angles to the same EF. And because EF is perpendicular both to А H

B GH and GK, EF is perpendicular a to the plane HGK passing

a 4. 11. through them: and EF is parallel

G to AB; therefore AB is at right

E

F angles b to the plane HGK. For

b 8. 11. the same reason, CD is likewise at right angles to the plane HGK.

с к

D Therefore AB, CD are each of them at right angles to the plane HGK. But if two straight lines be at right angles to the same plane, they shall be parallel < to one another. Therefore AB is c 6. 11. parallel to CD. Wherefore, two straight lines, &c. Q. E. D.

PROP. X. THEOR.

IF two straight lines meeting one another be paral. lel to two others that meet one another, and are not in the same plane with the first two, the first two and the other two shall contain equal angles.

Let the two straight lines AB, BC which meet one another be parallel to the two straight lines DE, EF that meet one another, and are not in the same plane with AB, BC. The angle ABC is equal to the angle DEF.

Take BA, BC, ED, EF all equal to one another; and join AD, CF, BE, AC, DF: because BA is equal and parallel to ED, there

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