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PRO P. 1.
NE part of a straight line cannot be in a plane and Sec N.
another part above it.
If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it : And since the straight line AB is in the plane, it can be
C produced in that plane : Let it be produced to D:And let any plane pass thro’the straight line AD, and be turned about it until it pass A B
D thro' the point C; and because the points B, C, are in this plane, the straight line BC is in it: a y. def. 1: Therefore there are two straight lines ABC, ABD in the same plane that have a common segment AB, which is impossible 6. b.Cor.11.1. Therefore one part, &c. Q. E. D.
Two straight lines which cut one another are in one see N.
plane, and three straight lines which meet one another are in one plane.
a 7. def. f.
Let two Araight lines AB, CD cut one another in E ; AB,
B in which EC, EB are, in the same are b
b. I. 11. CD, AB: Therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D.
IF two planes cut one another, their common section i:
a straight line.
Let two planes AB, BC, cut one another, and let the line
F lines DEB, DFB have the same extremi
ties, and therefore include a space be- C С a 10. Ax. I. twixt them; which is imposiblea: Therefore BD the common section of the planes
F a straight line stand at right angles to each of two
also be at right angles to the plane which paties through them, that is, to the plane in which they are.
Let the straight line EF stand at right angles to each of the straight lines AB, CD in E, the point of their intersection : EF is also at right angles to the plane passing through AB, CD.
Take the straight lines AE, EB, CE, ED all equal to one another; and through E draw, in the plane in which are AB,CD, any straight line GFH ; and join AD, CB ; then, from any point F in EF, draw FA, FG, FD, FC, FH, FB : And because the two straight lines, AE, ED are equal to the two BE, EC, and that they contain equal angles a AED, BEC, the base AD is equal b to the base BC, and the angle DAE to the angle EBC: And the angle AEG is equal to the angle BEH; therefore the triangles AEG, BEH have two angles of one equal to two angles of the other, each to each, and the sides AE, EB, adjacent to the equal angles, equal to one another; wherefore they sha! have their other lides equalo: GE is therefore
a 5. 1.
b 4. 1.
C 21. I.
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
b 4. I. equal o to the base FB ; for the same reason, CF is equal to FD: And because AD is equal to BC, and AF to FB, the two fides FA, AD are equal to the two FB, BC, each to each; and the base DF
F was proved equal to the base FC ; therefore the angle FAD is equal d to the angle FBC : Again, it was proved that GA is equal to BH, and also AF to FB; FA, then, and AG, are equal
C to FB and BH, and the angle FAG G bas been proved equal to the angle FBH; therefore the base GF is equal to the base FH : Again, because it
E H was proved, that GE is equal to and EF is common ; GE, EF are e
B qual to HE, EF; and the base GF is equal to the base FH; therefore the angle GEF is equal to the angle HEF; and consequently each of these angles is a right e angle. Therefore FE makes right angles with GH, e 10. def. I. that is, with any straight line drawn through E in 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 planer: Therefore EF is at right angles to the plane f 3. def. II. in which are AB, CD. Wherefore, if a straight line, &c. Q. E. D.
PRO P. V. THEO R.
IF three straight lines meet all in one point, and a See N.
straight line Itands at right angles to each of them in that point ; these three straight lines are in one and the
Let the straight line AB stand at right angles to each of the fraight lines BC, BD, BE, in B the
point where they meet ; BC, BD, BE are in one and the same plane.
It not, let, if it be pollible, 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
Book XI. are, shall be a straight a line ; let this be BF: Therefore the three m straight lines AB, BC, BF are all in one plane, viz. that which a 3. II.
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 14.11. to the plane pafling through them ; and therefore makes € 3. def. 11. right angles c with every straight
E line BC is not above the plane in which are BD and BE : Wherefore the three straight lines BC, BD, BE are in one and the same plane. Therefore, if three straight lines, &c. Q. E. D.
they shall be parallel to one another.
a def. 11.
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 fame plane; and make DE equal to AB, and join BÉ, AE, AD. Then, because A
C AB is perpendicular to the plane, it shall make right a 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,
D ABE is a right angle: For the same rea. fon, each of the angles CDB, CDE is a right angle: And because AB is equal to DE, and BD common, the two E fides AB, BD, are equal to the two ED, DB ; and they contain right 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, Book XI. in the triangles ABE, ÉDA, the base AE is common; there. W fore the angle ABE is equal to the angle EDA: But ABE is c 8. 1. a right angle; therefore EDA is also a right angle, 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 planed : But AB is in the plane in which are BD, DA, d 5. Il. because any three straight lines which meet one another are in one plane: Therefore AB, BD, DC are in one plane : And e 2. II. each of the angles ABD, BDC is a right angle; therefore AB is parallelf to CD. Wherefore, if two straight lines, &c. Q. E. D. f 28. 1.
I two straight lines be parallel, the straight line drawn See N.
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 poslable, above the plane, as EGF; and in the plane ABCD in which the
B parallels are, draw the straight line EHF from E to F, and since EGF also is a straight line, the
G two straight lines EHF, EGF
H Н include a space between them, wbich is impoflible. Therefore
a 10. Asia the straight line joining the C
F D points E, F is not above the plane in which the parallels AB, CD are, and is therefore in that plane. Wherefore, if two straight lines, &c. Q. E. D.
right angles to a plane; the other also Thall be at right angles to the same plane.