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Book XI. fore AD is a both equal and parallel to
and parallel to BE. Therefore AD and
C to BE. But straight lines that are paral
lel to the same straight line, and not in b. 9. II. the same plane with it, are parallel to
one another. Therefore AD is parallel C. 1. Ax. I. to CF; and it is equal < to it, and AC,
F and therefore a AC is equal and parallel
to DF. and because AB, BC are equal to DE, EF, and the base d. 8. I.
AC to the base DF; the angle ABC is equal d to the angle DEF.
10 draw a straight line perpendicular to a plane,
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
b. 11. I.
In the plane draw any straight line BC, and from the point A draw a AD perpendicular to BC. If then AD be also perpendiçular to the plane BH, the thing required is already done; but if it be not, from the point D draw
H C. 31. I. to DE; and thro' F draw GH pa
rallel to BC. and because BC is at
right angles to ED and DA, BC is d. 4. II. at right angles d to the plane pafsing thro' ED, DA. And GH is
B D parallel to BC; but if two straight lines be parallel, one of which is at right angles to a plane, the other shall be at right e angles to
the same plane; wherefore GH is at right angles to the plane thro' €. 3. Def. 11. ED, DA, and is perpendicular f to every straight line meeting it in
that planę. But AF, which is in the plane thro' ED, DA meets it.
e. 8. II.
therefore GH is perpendicular to AF, and consequently AF is Book XI. perpendicular to GH. 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.
10 ere& 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 an.
D B gles to the plane.
From any point B above the plane draw a BC perpendicular to it; and from A draw 6 AD parallel to BC. because therefore AD, CB are two parallel straight A c lines, and one of them BC is at right angles to the given plane, the other AD is also at right angles to it c. therefore a straight line has been erect- C. 8. II. ed at right angles to a given plane from a point given in it. Which was to be done.
2. II. II.
b. 31. 1.
PROM the same point in a given plane there cannot
be two straight lines at right angles to the plane, upon
the same side of it. and there can be but one per. pendicular to a plane from a point above the plane.
For, if it be possible, let the two straight lines AB, AC 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 thro' BA, AC; the common section of this with the given plane is a straight a line 2. 3. BIa
Book XI. paling through A. let DAE be their common section. therefore
the straight lines AB, AC, DAE are in one plane. and because
D А A E 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, b. 6. 11. they would be parallel to one another, which is absurd. There
fore from the same point, &c. Q. E. D.
LANES to which the same fraight line is perpen.
dicular, are parallel to one another.
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 meet; their common section shall be a
cause AB is perpendicular to the plane
H line BK which is in that plane. there
F fore ABK is a right angle. for the same
triangle ABK are equal to two right b. 19. s. angles, which is impöllible b. therefore
the planes CD, EF though produced do c. 8.Def.tr. not meet one another; that is, they are parallel c. Therefore
planes, &c. Q. E. D.
two straight lines meeting one another, be parallel See N.
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 pas. sing 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 fhall not meet though produced.
From the point B draw BG perpendicular to the plane which a. 11. it. passes through DE, EF, and let it meet that plane in G; and through G draw. GH parallel o to ED, and GK parallel to EF. b. 31. t. and because BG is perpendicular to the plane through DE, EF, it shall make right angles with every straight line meeting it in that plane c. but the straight B
F c.3.Def.11. lines GH, GK in that plane
К. meet it. therefore each of the angles BGH, BGK is a right angle. and because BA is paral- A lel d to GH (for each of them is
d. 9.11. parallel to DE, and they are not both in the same plane with it) the angles GBA, BGH are together equal o to two right angles. and BGH is a right angle, e. 29. 1. therefore also GBA is a right angle, and GB perpendicular to BA. for the fame 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 f. 4. 18. to the plane through 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 fame straight line is perpendicular, are parallel 8 to one another. 8. 14. 11. therefore the plane thro' AB, BC is parallel to the plane thro' DE, EF. Wherefore if two straight lines, &c. Q. E. D.
PROP. XVI." THEOR.
[F two parallel planes be cut by another plane, their
Let the parallel planes AB, CD be cut by the plane EFHG,
For, if it is not, EF, GH shall meet, if produced, either on the
H planes AB, CD produced meet
two straight lines be cut by parallel planes, they
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 cut by the plane EBDX, the common sections EX, BD are paral