equal to EH, and AG to BH. and because AE is equal to EB, and Book XI. FE common and at right angles to thein, the base AF is equal to b. 4. 1. the base FB; for the same reason CF is equal to FD. and because 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 base DF was proved equal to the T base FC; therefore the angle FAD is equal d to the angle FBC. again, it was d. $. . proved that AG is equal to BH, and also c AF to FB; FA then and AG, are equal to FB and BH, and the angle FAG has been proved equal to the angle TBH; therefore the base GF is equal to the E H 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 GEF is equal d to the angle HEF, and consequently each of these angles is a right e angle. Therefore FE makes right angles with GH, that €.10.Def.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 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 f. 3. Def.11. is at right angles to the plane in which are AB, CD. Wherefore if a straight line, &c. Q. E. D. 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 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, thall be a N a. 3. II. Book XI. straight a line; let this be BF. therefore the three straight lines AB, BC, BF are all in one plane, viz. that which passes thro' AB, BC. and because ABstands at rightangles toeach of the straight lines BD, b. 4. 11. BE, it is also at right angles to the plane passing thro' them; and there. A c. 3.Def.st. fore makes right angles with every straight line meeting it in tliat plane; F D B E the angle ABC, and they are both in the same plane, which is imposlibfe. therefore the straight 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 same plane. Therefore if three straight lines, &c. Q. E. D. IF two straight lines be at right angles to the fame 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 fame plane; and make DE equal to AB, and join BE, AE, AD. Then because AB is A perpendicular to the plane, it shall make a. 3.Def.11. right a angles with every straight line which meets it, and is in that plane. but 0. 4. 1. Book XI. c. 8. I. DA, and, in the triangles ABE, EDA, the base AE is common; therefore the angle ABE is equal to the angle EDA. but ABE is a right angle; therefore 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 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 plane e. therefore AB, e. 2. 11. 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. I, two straight lines, &c. Q. É. D. PROP. VII. THEOR. N N F two straight lines be parallel, the straight line drawn See N. from any point in the one to any point in the other is in the same plane with the parallels. IT Let AB, CD be parallel straight lines, and take any point E in the one, and the point Fin 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 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 EHF, EGF include a space betwixt them, which is impossible a. Therefore the straight line joining C FD 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. a. 10 Ax.i. (F two straight lines be parallel, and one of them is Sce N. at right angles to a plane ; the other also shall be at right angles to the fame plane. Book XI. с 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. 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 DE equal to AB, and join BE, AE, AD. And because AB is perpendicular to the plane, it is perpendicular to every straight 8. 3. Def.11. line which meets it, and is in that plane ! therefore each of the angles ABD, ABE, is a right angle. and because the straight line BD meets the parallel straight lines AB, CD, the angles ABD, b. 29. I. CDB are together equal b to two right angles. and ABD is a right EDB, because each of them is a right to the base BE. again, because AB is equal B angle; and therefore ED A is a right angle, is perpendicular e to the plane which passes thro' BD, DA, and . 3. Def.11. shall f make right angles with every straight line meeting it in that plane. but DC is in the plane passing thro' 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 angles to the plane passing thro' DE, DB, which is the same plane to which AB is at right angles. Therefore if two straight lines, &c. 0. E. D. e. 4. II. Book XI, PROP. IX. THEOR. wo straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another. TWO Let AB, CD be each of them parallel to EF, and not in the fame plane with it; AB shall be parallel to CD. In EF take any point G, from which draw, in the plane pasing thro' EF, AB, the straight line GH at right angles to EF; and in the plane passing thro' EF, CD, draw GK at right angles to the fame EF. and because EF is per. A H pendicular both to GH and GK, EF -B is perpendicular ? to the plane HGK 2. 4.. 12. passing thro' them. and EF is parallel G to AB; therefore AB is at right an E F gles o to the plane HGK. for the b. 8. 11. fame reason, CD is likewise at right angles to the plane HGK. therefore C K D AB, CD are each of them at right angles to the plane HGK. but if two straight lines be at right angles to the same plane, they shall be parallel to one another. therefore AB is parallel to CD. c. 6. 11. Wherefore two straight lines, &c. Q: E. D. IF. two straight lines meeting one another be parallel at meet one another, and are not in the same plane with the first two; the first two and the other two thall contain equal angles. to two Let the two ftraight lines AB, BC which meet one another be parallel to the two straight lines DE, EF that meet one another, and are not in the fame plane with AB, BC. The angle ABC is equal to the angle DEF. Take BA, BC, ED, EF all equal to one another; and join AD, CF, BE, AC, DF. because BA is equal and pasallel to ED, there |