Book XI. ON PROP. I. THEOR. NE part of a ftraight line cannot be in a plane and See N. another part above it. If it be poffible, let AB part of the straight line ABC be in the plane, and the part BC above it. and fince the straight line AB is in the plane, it can be produced in that planc. let it be produced to D. and let any plane pafs thro' the ftraight C line AD, and be turned about it until it pafs thro' the point C; and because A B D the points B, C are in this plane, the ftraight line BC is in it . therefore there are two ftraight lines a. 7. Def. 1ő ABC, ABD in the fame plane that have a common fegment AB, which is impoffible. Therefore one part, &c. Q. E. D. b.Cor.11.1. Tw PROP. II. THEOR. WO ftraight lines which cut one another are in See N. one plane, and three ftraight lines which meet. one another are in one plane. Let two ftraight lines AB, CD cut one another in E; AB, CD are in one plane. and three straight lines EC, CB, BE which meet one another, are in one plane. Let any plane pafs thro' the ftraight line A EB, and let the plane be turned about EB, produced if neceffary, until it pafs thro' the point C. then because the points E, C are in this plane, the ftraight line EC is in it. for the fame reason, the straight line BC is in the fame; and, by the Hypothefis, EB is in it. therefore the three ftraight lines EC, CB, BE are in one plane. but in the plane in which EC, EB are, in the fame are ↳ CD, b C E D B AB. therefore AB, CD are in one plane. Wherefore two ftraight lines, &c. Q. E. D a. 7. Def. 1. b. I. It. PROP. III. THE OR. two planes cut one another, their common fection is a ftraight line. Let two planes AB, BC cut one another, and let the line DB be their common fection; DB is a straight line. If it be not, from the point D to B draw in the plane AB the straight line DEB, and in the plane BC the straight B E line DFB. then two ftraight lines DEB, fore include a space betwixt them; which See N. a. 15. I. b. 4. I. C. 26. I. IF PROP. IV. THEOR. a ftraight line ftand at right angles to each of two ftraight lines in the point of their interfection, it fhall alfo be at right angles to the plane which paffes through them, that is, to the plane in which they are. Let the ftraight line EF ftand at right angles to each of the straight lines AB, CD in E the point of their interfection. EF is alfo at right angles to the plane paffing thro' AB, CD. Take the ftraight lines AE, EB, CE, ED all equal to one another; and thro' 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 that they contain equal angles AED, BEC, the bafe AD is equal 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 fides AE, EB, adjacent to the equal angles, equal to one another; wherefore they shall have their other fides equal . GE is therefore F b d. 9. f. equal to EH, and AG to BH. and becaufe AE is equal to EB, and Book XI. FE common and at right angles to them, the bafe AF is equal to the bafe FB; for the fame reafon CF is equal to FD. and be- b. 4. 1. caufe 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 bafe DF was proved equal to the bafe FC; therefore the angle FAD is equal to the angle FBC. again, it was proved that AG is equal to BH, and alfo AF to FB; FA then and AG, are equalA to FB and BH, and the angle FAG has, been proved equal to the angle FBH; therefore the bafe GF is equal to the bafe FH. again, because it was proved that GE is equal to EH, and EF is com-mon; GE, EF are equal to HE, EF; and C H B the bafe GF is equal to the bafe FH; therefore the angle GEF is equal to the angle HEF, and confequently each of thefe angles is a right angle. Therefore FE makes right angles with GH, that e.ro.Def.t. is, with any ftraight line drawn thro' E in the plane paffing thro' AB, CD. In like manner it may be proved that FE makes right angles with every ftraight line which meets it in that plane. But a ftright line is at right angles to a plane when it makes right angles, with every ftraight line which meets it in that plane f. therefore EF 1.3 Def if. is at right angles to the plane in which are AB, CD. Wherefore if a ftraight line, &c. Q. E. D. IF PROP. V. THEOR.' F three ftraight lines meet all in one point, and a See . ftraight line ftands at right angles to each of them in that point; thefe three ftraight lines are in one and the fame plane. Let the ftraight line AB ftand at right angles to each of the ftraight lines BC, BD, BE, in B the point where they meet; BC, BD, BE are in one and the fame plane. If not, let, if it be poffible, BD and BE be in one plane, and BC be above it; and let a plane pass through AB, BC, the common fection of which with the plane, in which BD and BE are, shall be N Book XI. ftraight line; let this be BF. therefore the three straight lines AB, BC, BF are all in one plane, viz. that which paffes thro' AB, BC. and becaufe AB ftands at right angles to each of the ftraight lines BD, BE, it is alfo at right angles b to the a. 3. 11. b. 4. 11. plane paffing thro' them; and there-A e.3.Def.11. fore makes right angles with every F B ftraight line meeting it in that plane; PROP. VI. THEOR. E F two ftraight lines be at right angles to the fame plane, they fhall be parallel to one another. Let the ftraight lines AB, CD be at right angles to the fame plane; AB is parallel to CD. 10 Let them meet the plane in the points B, D, and draw the ftraight line BD, to which draw DE at right angles, in the fame plane; and make DE equal to AB, and join BE, AE, AD. Then because AB is perpendicular to the plane, it fhall make 3.Def.11. right angles with every ftraight 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 B B. 4. I. a angles ABD, ABE is a right angle. for b D E fides AB, BD, are equal to the two ED,DB; and they contain right angles; therefore the bafe AD is equal to the bafe BE. again, becaufe AB is equal to DE, and BE to AD; AB, BE are equal to ED, C DA, and, in the triangles ABE, EDA, the bafe AE is common; Book XI. therefore the angle ABE is equal to the angle EDA. but ABE is a right angle; therefore EDA is alfo a right angle, and ED per- c. 8. 1. pendicular to DA. but it is alfo perpendicular to each of the two BD, DC. wherefore ED is at right angles to each of the three ftraight lines BD, DA, DC in the point in which they meet. therefore these three ftraight lines are all in the fame plane 4. but AB d. s. 11. is in the plane in which are BD, DA, because any three ftraight lines which meet one another are in one plane. therefore AB, BD, DC are in one plane. and each of the angles ABD, BDC is a right angle; therefore AB is parallel to CD. Wherefore if f. 28. 1. two ftraight lines, &c. Q. E. D. f PROP. VII. THEO R. C. 2. II. IF two straight lines be parallel, the ftraight line drawn See N. from any point in the one to any point in the other is in the fame plane with the parallels. Let AB, CD be parallel ftraight 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 fame plane with the parallels. If not, let it be, if poffible, above the plane, as EGF; and in the plane ABCD in which the paral lels are, draw the ftraight line EHF from E to F; and fince EGF alfo is a ftraight line, the two ftraight lines EHF,EGFinclude a fpace betwixt them, which is impoffible". Therefore the straight line joining the points E, F is not above the A E B G H a.ro. Az. 1. C D plane in which the parallels AB, CD are, and is therefore in that plane. Wherefore if two ftraight lines, &c. Q. E. D. PROP. VIII. THEOR. IF two ftraight lines be parallel, and one of them is See at right angles to a plane; the other alfo fhall be at right angles to the fame plane. |