Book XI. PROP. I. THE OR. ONE part of a ftraight line cannot be in a plane and see N. 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 produced in that planc. let it be produced to D. and let any plane pass thro' the straight line AD, and be torned about it until A А B D it pass thro’the point C; and because the points B, C are in this plane, the straight line BC is in it. therefore there are two straight lines a. 7. Def. så ABC, ABD in the same plane that have a common segment AB, which is imposible b. Therefore one part, &c. Q. E. D. b.Cor.11.1. PRO P. II. THEOR. wo straight lines which cut one another are in Sec N. one plane, and three straight lines which meet one another are in one plane. TW Let two straight 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 pass thro’ the straight line A D EB, and let the plane be turned about EB, produced if necessary, until it pass thro' the point C. then because the points E, C are a. 7. Def.z. in this plane, the straight line EC is in it. for the same reason, the straight line BC is in the fame; and, by the Hypothesis, EB is in it. therefore the three straight lines EC, CB, BE are in one plane. but in the plane B in which EC, EB are, in the same are 6 CD, AB. therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D b.i. it. Book XI. PRO P. III. THEOR. See N. T two planes cut one another, their common section is a straight line. Let two planes AB, BC cut one another, and let the line DB B В F fore include a space betwixt them; which D A fection of the planes AB, BC cannot but be a straight ime. Wherefore if two pianes, &c. Q. E. D. PROP. IV. IV. THE OR. See Y. TF 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 interfection. EF is also at right angles to the plane passing thro’ AB, CD. Take the straight lines AE, EB, CE, ED all equal to one another; and thro' E draw, in the plane in vihich 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 base AD is equal b to the base BC, and the angle DAE to the angłe 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 sides equal C. GE is therefore a. 15.1. b. 4. 1. c. 26. 1. equal to EH, and AG to BH. and because AE is equal to EB, and Book XI. F d. 9. proved that AG is equal to BH, and also 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 base GF is equal H base FH. again, because it was proved that GE is equal to EH, and EF is com- B mon; GE, EP are equal to HE, EF; and the base GF is equal to the base FH; therefore the angle GEF is equal a to the angle HEF, and consequently each of these angles is a right angle. Therefore FE makes right angles with CH, that c.io.Define is, with any straight line drawn thro' E in the plane passing thro' AB, CD. In like manner it may be proved that FE makes righe 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 1,3 Der it, is at right angles to the plane in which are AB, CD. Wherefore if a straight line, &c. Q. E. D. to the PRO P. V. THEOR. a 1 straight line stands at right angles to each of them in that point ; these three straight lines are in one and tlie samé plane. Let the straight line AB ftand at right angles to each of the Atraight lines DC, BD, BE, in B the point where they meet; BC, BD, BE are in one and the same plane. If not, lct, 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, fall be t N a. 3. II. b. 4. II. Book XI. straight · line; let this be BF. therefore the three straight lines AB, CBC, EF are all in one plane, viz. that which passes thro' AB, BC. and because AB stands at right angles to each of the straight lines BD, BE, it is also at right angles b to the plane passing thro' them; and there- A 6.3.Def.11. fore makes right angles with every straight line meeting it in that plane; IF PRO P. VI. THEOR. plane, they shall be parallel to one another. 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 BE, AE, AD. Then because AB is A perpendicular to the plane, it shall make 2:3.Defar. right angles with every straight line which meets it, and is in that plane. but D E Gides AB, BD, are equal to the two ED,DB; and they contain right b. 4. 1. angles; therefore the base AD is cqual to the base BE. again, becauss 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 common ; Book XI. therefore the angle ABE is equal to the angle EDA. but ABE M is a right angle; therefore EDA is also a right angle, and ED per. c. 8. 1. pendicular 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 d. 5. 11, is in the plane in which are BD, DA, because any three straight lines which meet one another are in one planee. 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 straight lines, &c. Q. E. D. e. 2. II, IF iwo straight lines be parallel, the straight line drawn See M. 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 fame plane with the parallels. If not, let it be, if possible, above the plane, 'as EGF; and in the planc ABCD in which the parallels are, draw the straight line EHF A E B from E to F; and since EGF also is a straight line, the two straight H lines EHT,EGFinclude a space betwixt them, which is impossible". Therefore the straight line joining C D the points E, F is not above the plane in which the parallels AB, CD are, and is therefore in that plane. Wherefore if iwo straight lines, &c. Q. E. D. 2.10.Az. PROP. Vill. THE OR. IF at right angles to a plane; the other alio fhall be at right angles to the same plane. |