Book XI. b. 29. 1. Let AB, CD be two parallel ftraight lines, and let one of them AB be at right angles to a plane; the other CD is at right angies to the fame 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 2.3.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 ftraight lines AB, CD, the angles ABD, CDB are together equal to two right angles. and ABD is a right angle; therefore alfo CDB is a right angle, and CD perpendicular to BD. and because AB is equal to DE, and BD common, the two AB, BD, are equal to the two ED, DB, and the angle ABD is equal to the angle A EDB, because each of them is a right angle; therefore the bafe AD is equal to the bafe BE. again, becaufe AB is equal to DE, and BE to AD; the two AB, BE are equal to the two ED, DA; and the bafe AE is common to the triangles ABE, B EDA; wherefore the angle ABE is equal to the angle EDA. and ABE is a right angle; and therefore EDA is a right angle, and ED perpendicular to DA. but it is alfo perpendicular to BD; therefore ED C. 4. 1. d. 8. .. C. 4. 11. e E is perpendicular to the plane which paffes thro' BD, DA, and £3.Def.11. fhall make right angles with every ftraight line meeting it in that plane. but DC is in the plane paffing 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 alfo at right angles to DB; CD then is at right angles to the two ftraight lines DE, DB in the point of their interfection D; and therefore is at right angles to the plane paffing thro' DE, DB, which is the fame plane to which AB is at right angles. Therefore if two ftraight lines, &c. Q. E. D. PROP. IX. THEOR. WO ftraight lines which are each of them parallel TWO to the fame ftraight line, and not in the fame plane with it, are parallel to one another. Let AB, CD be each of them parallel to EF, and not in the fame plane with it; AB fhall be parallel to CD. In EF take any point G, from which draw, in the plane paffing thro' EF, AB, the ftraight line GH at right angles to EF; and in the plane paffing thro' EF, CD, draw GK at right angles to the fame LF. and becaufe EF is per- A H pendicular both to GH and GK, EF is perpendicular to the plane HGK paffing thro' them. and EF is parallel 197 Book XI. a. 4. 11. G to AB; therefore AB is at right an E F gles to the plane HGK. for the D fame reafon, CD is likewife at right angles to the plane HGK. therefore CK b. 8. 11. AB, CD are each of them at right angles to the plane HGK. but if two ftraight lines be at right angles to the fame plane, they fhall be parallel to one another. therefore AB is parallel to CD. c. 6. 11, Wherefore two straight lines, &c. Q. E. D. PROP. X. THEOR. IF two ftraight lines meeting one another be parallel to two others that meet one another, and are not in the fame plane with the first two; the first two and the other two fhall contain equal angles. Let the two straight 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 parallel to ED, there, Book XI. fore AD is both equal and parallel to b. 9. 11. d. 8. 1. 2 12. 1. b. 11. 1. C. 31. I. d. 4. 11. c. 8. 11. C d B E F to DF. and becaufe AB, BC are equal to DE, EF, and the base AC to the bafe DF; the angle ABC is equal to the angle DEF. Therefore if two ftraight lines, &c. Q. E. D. PROP. XI. PROB. O draw a ftraight line perpendicular to a plane, from a given point above it. Let A be the given point above the plane BII; it is required to draw from the point A a ftraight line perpendicular to the plane EH. In the plane draw any ftraight line BC, and from the point A draw AD perpendicular to BC. If then AD be allo perpendicular to the plane DH, the thing required is already done, but if it be not, from the point D draw b c in the plane BH, the ftraight line B D parallel to BC; but if two ftraight lines be parallel, one of which is at right angles to a plane, the other fhall be at right angles to the fame plane; wherefore CH is at right angles to the plane thro' f. 3. Def. 11. ED, DA, and is perpendicular f to every ftraight line meeting it in that plane. But AF, which is in the plane thro' ED, DA meets it. therefore GH is perpendicular to AF, and confequently AF is Bock Xi. perpendicular to GH. and AF is perpendicular to DE; therefore AF is perpendicular to each of the ftraight lines GH, DE. but if a straight line ftands 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 paffing through them. but the plane paffing through ED, GH is the plane BH; therefore AF is perpendicular to the plane BH. therefore from the given point A above the plane BH, the ftraight line AF is drawn perpendicular to that plane. Which was to be done. T PROP. XII. PRO B. O erect a ftraight line at right angles to a given Let A be the point given in the plane; it is required to erect a ftraight line from the point A at right an gles to the plane. From any point B above the plane draw BC perpendicular to it; and from A drawb AD parallel to BC. because therefore AD, CB are two parall straight lines, and one of them BC is at right an gles to the given plane, the other AD isl D a. II. II. b. 31. 1. A alfo at right angles to it . therefore a ftraight line has been erect- c. 8. 11. ed at right angles to a given plane from a point given in it. Which was to be done. FR 'ROM the fame point in a given plane there cannot be two ftraight lines at right angles to the plane, upon the fame fide of it. and there can be but one per pendicular to a plane from a point above the plane. For, if it be poffible, let the two ftraight lines AB, AC be at right angles to a given plane from the fame point A in the plane, and upon the fame fide of it; and let a plane pafs thro' BA, AC; the common fection of this with the given plane is a ftraight line a Book XI. paffing through A. let DAE be their common fection. therefore the ftraight lines AB, AC, DAE are in one plane. and because CA is at right angles to the given plane, it shall make right angles b. 6. II. with every ftraight line meeting it in which is impoffible. Alfo, from a point above a plane there can There be but one perpendicular to that plane; for if there could be two, PLANES LANES to which the fame ftraight line is perpendicular, are parallel to one another. Let the ftraight line AB be perpendicular to each of the planes CD, EF; thefe planes are parallel to one another. If not, they shall meet one onother when produced; let them meet; their common fection fhall be a ftraight line GH, in which take any .3. Def. 11. EF, it is perpendicular to the ftraight C K H F A B E D Therefore c.8.Def.11. not meet one another; that is, they are parallel planes, &c. Q. E. D. |