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PROP. III. THEOR.
IF two planes cut one another, their common sec. tion is a straight line.
Let two planes AB, BC, cut one another, and let the line DB
F a 10. Ax. is impossible a: therefore BD the common 1. section of the planes AB, BC cannot but
A be a straight line. Wherefore, if two
D planes, &c. Q. E. D.
PROP. IV. THEOR.
IF a straight line stand 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 which passes 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 GEH; 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 a 15. 1. that they contain equal angles a AED, BEC, the base AD is b 4. 1. equal b to the base BC, and the angie DAE to the angle EBC:
and the angle AEG is equal to the angle BEH a ; 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,
aujacent to the equal angles, equal to one another; wherec 26. 1. fore they shall have their other sides equalc: GE is there fore
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 in 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,
с 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
H to EH, and EF is common ; CE, EF are equal to HE, EF ; and the base GF is
B 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 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. ED.
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
v 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 Α,
in that plane meets it: therefore the
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
angles with every straight line which meets
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:n 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.
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
H tween them, which is impossible a
a 10, 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.
PROP. VIII. THEOR.
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.
Book XI. 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
pendicular to the plane, it is perpendicular to every straight line a 3. which meets it, and is in that planea : 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
A angle EDB, because each of them is a
right angle; therefore the base AD is c4 1.
equal < to the base BE: again, because AB
D d 8. 1.
ABE is equald to the angle EDA: and
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 shallf make right an11.
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 intersection D; and therefore is at right anglese to the plane passing through DE, DB, which is the same plane to which AB is at right angles. Therefore, if two straight lines, &c. Q. E. D.