Book XI. PROP. I. THEOR. ONE part of a straight line cannot be in a plane and see N. another part above it. С 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 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 thro' the point C; and А B D because the points B, C are in this plane, the straight line BC is in it". therefore there are two straight a. 7. Del. s. lines ABC, ABD in the same plane that have a common segment AB, which is impossible b. Therefore one part, &c. Q. E, D. b.Cor.11., PROP. II. THEOR. TWO plane, and three straight lines which meet one another are in one plane. 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 through the straight AL line EB, and let the plane be turned about EB, produced if necessary, until it pass through the point C. then because the E points E, C are in this plane, the straight line EC is in it". for the same reason, the 2. 7. Def.si 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. B but in the plane in which EC, EB are, in the same are o CD, 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 section is a straight line. Let two planes AB, BC cut one another, and let the line DB be B E F с A D See N. PRO P. IV. THEOR. straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them, tharis, 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 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 Fin 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 a. 15. 1. equal angles a AED, BEC, the base AD is equal b to the base BC, b. 4. I. 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 ; 6. 26.1. wherefore they shall have their other sides equal . GE is theretore equa! d. 8. equal to EH, and AG to BH. and because AE is equal to EB, and Book XI. FE common and at right angles to them, the base AF is equal to the base FB; for the same reason CF is equal to FD. and because b. 4.1. 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 proved equal to the base FC; therefore the angle FAD, is eequal o to the angle FBC. again, it was proved that AG is equal to BH, and also A AF to FB; FA then and AG, are equal to FB and BH, and the angle FAG has G been proved equal to the angle FBH ; therefore the base GF is equal to the E H Н base FH. again, because it was proved that GE is equal to EH, and EF is com B mon; GE, EF are equal to HE, EF ; and the base GF is equal to the base FH; therefore the angle CEF iz equal d to the angle HEF, and consequently each of these angles is a right e angle. Therefore FE makes right angles with CH, that e. 10. Déf.s. 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 righi angles with every straight line which meets it in that plane. But à straight line is at right angles to a plane when it makes right angles with every straight line which meets it in that planef, therefore EF 4.3.Def.ne. is at right angles to the plane in which are AB, CD. Wherefore if a straight line, &c. Q. E. D. PROP. V. PROP. V. THEOR. IF F 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 fame plane. Let the straight line AB seand 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 fame 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 fe tion of which with the plane, in which BD and BE are, shall N be Book X1. be a straight a 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 AB stands at right angles to each of the b.4. 11. straight lines BD, BE, it is also at right angles b to the plane passing through them; and therefore makes c.3.Def. 11. right angles « with every straight line A meeting it in that plane; but BF which c F D B 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 BE, AE, с A every straight line which meets it, and is in D E common, the two sides AB, BD, are equal to the two ED, DB; and they contain right angles; therefore the #41. base AD is equal to the base BE. again, because 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 planet. but AB is in the plane in which are BD, d. 5. 17. 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 c. 2. 1 si the angles ABD, BDC is a right angle; therefore AB is parallel é f. 28. 1. to CD. Wherefore if two straight lines, &c. Q. E. D. PRO P. VII. THEO R. IF two straight lines be parallel, the straight line drawn Set 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 ini the one, and the point F in the other. the straight line which joins É 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 E le!s are, draw the straight line A B. EHF from E to F; and since EGF also is a straight line, the two G H Н straight lines EHF, EGF include a space betwixt them, which is impossible. Therefore the straight C F D 2.10. Ale line joining the 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. PROP. VIII. THEOR. IF two straight lines be parallel, and one of them is at scé 7. right angles to a plane ; the other also shall be at right angles to the same plane. |