Ο PROP. I. THEOR. Book XI. 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 ftraight 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 plane. let it be produced to D. and let any plane pafs thro' the straight line AD, and be turned about it until it pass thro' the point C; and because the points B, C are in this A B D C plane, the straight line BC is in it. therefore there are two straight a. 7. Def. 1. lines ABC, ABD in the fame plane that have a common segment AB, which is impoffible b. Therefore one part, &c. Q. E. D. TWO PROP. II. THEOR. b. Cor.11.1. 'WO straight lines which cut one another are in one See N. plane, and three ftraight lines which meet one ano ther are in one plane. Let two straight lines AB, CD cut one another in E; AB, CD are in one plane. and three ftraight lines EC, CB, BE which meet one another, are in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced if neceffary, until it pafs through the point C. then because the points E, C are in this plane, the straight line EC is in it. for the fame reason, the ftraight 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 A D E 2. 7. Def. 1. B. the fame are bCD, AB. therefore AB, CD are in one plane. Where- b. 1. 11. fore two straight lines, &c. Q. E. D. PROP. Book XI. PROP. III. THEOR. See N. IF 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 B E F C A D DEB, and in the plane BC the straight line DFB. then two straight lines DEB, DFB have the fame extremities, and therefore include a space betwixt them; which is im10. Ax. 1. poffible. therefore BD the common fection of the planes AB, BC cannot but be a ftraight line. Wherefore if two planes, &c. Q. E. D. Sce N. IF a Fa 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 ftraight 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 straight 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 ftraight lines AE, ED are equal to the two BE, EC, and that they contain 2. 15. 1. equal angles a AED, BEC, the bafe AD is equal to the bafe 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; €. 26. 1. wherefore they shall have their other fides equal. GE is therefore b. 4. I. equal equal to EH, and AG to BH. and because AE is equal to EB, and Book XI. to the two FB, BC, each to each; and b G d. 8. 1. C E H B and the bafe GF is equal to the bafe FH; therefore the angle CEF is equal to the angle HEF, and confequently each of these angles is d a right angle. Therefore FE makes right angles with GH, that c. 10.Def.1. is, with any straight 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 ftraight line is at right angles to a plane when it makes right angles with every straight line which meets it in that plane f. is at right angles to the plane in which are AB, CD. a straight line, &c. Q. E. D. IF therefore EFf. 3.Def. 11. Wherefore if three straight lines meet all in one point, and a sœ N. 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 pafs through AB, BC, the common festion of which with the plane, in which BD and BE are, shall N be a Book XI. be a ftraight line; let this be BF. therefore the three straight lines AB, BC, BF are all in one plane, viz. that which passes through a 3.11. AB, BC. and because A B stands at right angles to each of the II. b. 4. 11. ftraight lines BD, BE, it is alfo at right angles to the plane paffing through them; and therefore makes e.3.Def. 11. right angles with every straight line the angle ABC, by the Hypothesis, is is impoffible. therefore the straight C F -D 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 fame plane. Therefore if three straight lines, &c. Q. E. D. IF PROP. VI. THEOR. two straight 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. 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 BE, AE, AD. then because AB is perpendicular to A a.g. Def..the plane, it fhall make right angles with a right angle. for the fame reason, each of common, the two fides AB, BD, are equal E C D to the two ED, DB; and they contain right angles; therefore the 1. bafe AD is equal to the bafe BE. again, becaufe AB is equal to DE, DE, and BE to AD; AB, BE are equal to ED, DA, and, in the Book XI. triangles ABE, EDA, the base AE is common; therefore the angle ABE is equal to the angle EDA. but ABE is a right angle; there- c. 8. 1. fore 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 fame plane. but AB is in the plane in which are BD, d. s. iz. DA, because any three straight lines which meet one another are in one plane. therefore AB, BD, DC are in one plane. and each of e. 2. 11. the angles ABD, BDC is a right angle; therefore AB is parallel f f. 18. 1. to CD. Wherefore if two straight lines, &c. Q. E. D. IF F PRO P. VII. THEOR. two ftraight 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 straight lines, and take any point E in the one, and the point F in the other. the straight line which joins É and F is in the fame plane with the parallels. A If not, let it be, if poffible, above the plane, as EGF; and in the plane ABCD in which the parallels are, draw the ftraight line EHF from E to F; and fince EGF alfo is a ftraight line, the two ftraight lines EHF, EGF include a fpace betwixt them, which is impoffible. Therefore the straight C line joining the points E, F is not E B G H F D above the plane in which the parallels AB, CD are, and is therefore in that plane. Wherefore if two ftraight lines, &c. Q. E. D. IF PROP. VIII. THEOR. F two ftraight lines be parallel, and one of them is at set 17, right angles to a plane; the other alfo fhall be at right angles to the fame plane. |