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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.

D

с к 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

[blocks in formation]

Book XI. fore AD is a both equal and parallel to BE.

For the same reason CF is equal and paa 33. 1. rallel to BE. Therefore AD and CF are

each of them equal and parallel to BE.
But straight lines that are parallel to the

same straight line, and not in the same b 9. 11. plane with it, are parallel b to one ano

ther. Therefore AD is parallel to CF; cl. Ax.l. and it is equal c to it, and AC, DF join

them towards the same parts; and there-
fore a AC is equal and parallel to DF.
And because AB, BC are equal to DE,

EF, and the base AC to the base DF; the d 8. 1.

angle ABC is equal 4 to the angle DEF.
straight lines, &c. Q.E.D.

[blocks in formation]

PROP. XI. PROB.

TO draw a straight line perpendicular to a planc, from a given point above it.

Let A be the given point above the plane BH; it is required to draw from the point A a straight line perpendicular to the plane BH.

In the plane draw any straight line BC, and from the point A a 12. 1. draw a AD perpendicular to BC. If then AD be also perpendi

cular to the plane BH, the thing required is already done; but if b 11. 1. it be not, from the point D drawb, in the plane BH, the straight

line DE at right angles to BC: and from the point A draw AF c 31. 1. perpendicular to DE ; and through F drawcGH parallel to BC: and because BC is at right angles

A
to ED and DA, BC is at right an-
d 4. 11. gles d to the plane passing through

E
ED, DA. And GH is parallel to
BC; but, if two straight lines be

F
G

H parallel, one of which is at right

angles to a plane, the other shall e 8. 11. be at right e angles to the saine

plane; wherefore GH is at right
angles to the plane through ED,

B

D

с f3. def. DA, and is perpendicular f to 11.

every straight line meeting it in that plane. But AF, which is in the plane through ED, DA, meets it: therefore GH is per

pendicular to AF; and consequently AF is perpendicular to GH; Book XI. and AF is perpendicular to DE: therefore AF is perpendicular to each of the straight lines GH, DE. But if a straight line stands at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane passing through them. But the plane passing through ED, GH is the plane BH ; therefore AF is perpendicular to the plane BH; therefore, from the given point A, above the plane BH, the straight line AF is drawn perpendicular to that plane. Which was to be done.

PROP. XII. PROB.

TO erect a straight line at right angles to a given plane, from a point given in the plane.

Let A be the point given in the plane ; it is required to erect a straight line from the point A at right angles to the plane.

D B From any point B above the plane drawa

a 11. 11. BC perpendicular to it; and from A draw b

b 31. 1. AD parallel to BC. Because, therefore, AD, CB are two parallel straight lines, and one of them BC is at right angles to the given plane, the other AD is also at

A

с right angles to its. Therefore a straight

C 8, 11, line has been erected at right angles to a given plane from a point given in it. Which was to be done.

PROP. XIII. THEOR.

FROM the same point in a given plane, there can. not be two straight lines at right angles to the plane, upon the same side of it; and there can be but one perpendicular to a plane from a point above the plane.

For, if it be possible, let the two straight lines AC, AB be at right angles to a given plane from the same point A in the plane, and upon the same side of it; and let a plane pass through BA,

Book XI. AC; the common section ofthis with the given plane is a straight

line passing through A: let DAE be their common section: therea 3. 11. fore the straight lines AB, AC, DAE are in one plane: and be

cause CA is at right angles to the given plane, it shall make
right angles with every straight line meeting it in that plane.
But DAE, which is in that plane, meets CA, therefore CAE is
a right angle. For the same reason
BAE is a right angle. Wherefore

B

C
the angle CAE is equal to the angle
BAE; and they are in one plane,
which is impossible. Also, from a
point above a plane, there can be
but one perpendicular to that plane;
for, if there could be two, they would

D
A

E b 6.11. be parallel b to one another, which

is absurd. Therefore, from the same point, &c. Q. E. D.

PROP. XIV. THEOR.

PLANES to which the same straight line is

per pendicular, are parallel to one another.

K

Let the straight line AB be perpendicular to each of the planes CD, EF; these planes are parallel to one another.

If not, they shall meet one another when produced ; let them mect; their common section shall be a straight line GH, in which take any

G point K, and join AK, BK : then, be

cause AB is perpendicular to the plane a 3. def. EF, it is perpendicular a to the straight 11. line BK which is in that plane. There

C fore ABK is a right angle. For the

н same reason, BAK is a right angle ;

F wherefore the two angles ABK, BAK

A

B of the triangle ABK are equal to two b 17. 1. right angles, which is impossible b: therefore the planes CD, EF, though

E produced, do not meet one another; . 8. def. that is, they are parallel c. Therefore,

D 11.

planes, &c. Q. E. D.

.

Book XI.

PROP. XV. THEOR.

IF two straight lines meeting one another, be pa- See N. rallel to 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.

G

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, though produced.

From the point B draw BG perpendicular a to the plane a 11. 11. which passes 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. 1. rallel to EF: and because BG is perpendicular to the plane through DE, EF, it shall

E make right angles with every straight line meeting it in that

F plane c. But the straight lines B

c 3. def. GH, GK in that plane meet

с

K 11. it : therefore each of the angles BGH, BGK is a right angle: and because BA is pa

D rallel d to GH (for each of A

d 9.11. them is parallel 10 DE, and

H Я they are not both in the same plane with it) the angles GBA, BGH are together cquale to iwo right angles: and BGH is a e 29. 1. right angle; therefore 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 perpendicularf to the plane through BA, BC ; and it is f 4. 11. 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 AB, 8 14. 11., BC is parallel to the plane through DE, EF.' Wherefore, if two straight lines, &c. Q. E. D.

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