4. 1. • 8. 1. 4. 1. 端 * * equal to EH, and AG to BH: and because AE is equal to EB, and 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 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 equal to the angle FBC: again, it was proved that GA is equal to BH, and also AF to FB; therefore FA and AG, are equal to FB and BH, each to each; and the angle FAG has been proved equal to the angle FBH; therefore the base GF is equal to the base FH: again, because it was proved that GE is equal to EH, and EF is common, therefore GE, EF are equal to HE, EF, each to each; and the base GF is equal to the base FH; therefore the angle GEF is equal * to the angle HEF; and consequently each of these angles is a * 10 Def. 1. right angle: therefore FE makes right angles with GH, that is, with any straight line drawn through E, in the plane passing through 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 * 3 Def. 11. which meets it in that plane*: therefore EF is at right angles to the plane in which a. e AB, CD. Wherefore, if a straight line, &c. Q. E. D. 8. 1. * See N. * 3. 11. PROPOSITION V. THEOR.-If three straight lines meet all in one point, and a straight line stand 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 shall be in one and the same plane. B -D E 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, is a straight* line; let this be BF: therefore the three straight lines AB, BC, BF are all in one plane, viz. that which passes through AB, BC: and * . 4. 11. 3 Def. 11. because AB stands at right angles to each of the straight lines BD, BE, it is also at right angles to the plane passing through them, and therefore makes right angles * with every straight line in that plane which meets it: but BF, which is in that plane, meets it; therefore the angle ABF is a right angle: but the angle ABC, by the hypothesis, is also a right angle; therefore the angle ABF is equal to the angle ABC, and they are both in the same plane; which is † †9 Ax. impossible: 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. PROPOSITION 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 shall be parallel to CD. AN C * 3 Def. 11. 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 11. I. same plane; and make † DE equal to AB, and join BE, AE, † 1. 3. AD. Then, because AB is perpendicular to the plane, it shall make right angles with every straight line. which meets it, and is in that plane: but BD, BE, which are in that plane, do each of them meet AB; therefore each of the angles ABD, ABE is a right angle; for the same reason, each of the angles CDB, CDE is a right angle: and because AB is equal to DE, and BD com B mon, the two sides AB, BD are equal to the two ED, DB, each to each; and they contain right angles; therefore the base AD is equal to the base BE: again, because AB is equal to DE, * 4. 1. and BE to AD; AB, BE are equal to ED, DA, each to each; and, in the triangles ABE, EDA, the base AE is common; therefore the angle ABE is equal to the angle EDA: but ABE 8.1. 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 5. 11. * 2. 11. * 28. 1. same plane: but AB is in the plane in which are BD, 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 the angles ABD, BDC is a right angle; therefore AB is parallel to CD. Wherefore, if two straight lines, &c. Q. E. D. * See N. PROPOSITION VII. THEOR.-If two straight lines be parallel, the straight line drawn 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, shall be in the same plane with the parallels. A E H B G 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 from E to F: and since EGF also is a straight line, the two straight lines * 10 Ax. 1. EHF, EGF include a space between them; which is ble: therefore the straight 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. impossi See N. * 7. 11. † 11. 1. † 3. 1. PROPOSITION VIII. THEOR.-If two straight lines be parallel, and one of them is at right angles to a plane, the other also shall be at right angles to the same plane. Let AB, CD be two parallel straight lines, and let one of them AB be at right angles to a plane: the other CD shall be 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 * 3 Def. 11. to every straight line which meets it and is in that plane* ; -e each of the angles ABD, ABE is a right angle: and o right angles: and ABD is a right E 29. 1. D 4. 1. CD perp dicular to BD: and because AB is equal to DE, and BD common, the two AB, BD B are equal to the two ED, DB, each to each; and the angle ABD is equal to the angle EDB, because each of them is a right angle; therefore the base AD is equal * o the base BE: again, because AB is equal to DE, and BE to AD, the two AB, BE are equal to the two ED, DA, each to each; and the base AE is common to the triangles ABE, EDA; wherefore the angle ABE is equal to the angle EDA: but 8. 1. ABE is a right angle; and therefore EDA is a right angle, and ED perpendicular to DA: but it is also perpendicular to † BD; + Constr. therefore ED is perpendicular to the plane which passes through BD, DA; and therefore makes right angles with every straight line meeting it in that plane: but DC is in the plane passing through 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; therefore CD 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 4. 11. plane passing through DE, DB, which is the same plane to which AB is at right angles. Therefore, if two straight lines, &c. Q. E. D. 4. 11. 3 Def. 11. PROPOSITION IX. THEOR. Two 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. Let AB, CD be each of them parallel to EF, and not in the same plane with it: AB shall be parallel to CD. B E G F CK D In EF, take any point G, from which t draw, in the plane passing through EF, AB, the straight line GH at right angles to EF; and in the plane passing through EF, CD, draw GK at right angles to the same EF. And because EF N 4. 11. * S. 11. * 6. 11. * is perpendicular both to GH and GK, EF is perpendicular * to the plane HGK passing through them and EF is parallel to AB; therefore AB is at right angles to the plane HGK: for the same reason, CD is likewise at right angles to the plane HGK; therefore AB, CD are each of them at right angles to the plane HGK. But if two straight lines are at right angles to the same plane, they are parallel to one another; therefore AB is parallel to CD. Wherefore, two straight lines, &c. Q. E. D. * 38. 1. * 9.11. * 1 Ax. 1. * 33. 1. * 8. 1. PROPOSITION X. THEOR.-If two straight lines meeting one another be parallel to two others that meet one another, and are not in the same plane with the first two, the first two and the other two shall 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 same plane with AB, BC: the angle ABC shall be equal to the angle DEF. D B C. Take BA, BC, ED, EF all equal to one another; and join AD, CF, BE, AC, DF: then because BA is equal and parallel to ED, therefore Af AD is both equal and parallel to BE: for the same reason, CF is equal and parallel to BE; therefore AD and CF are each of them equal and parallel to BE. But straight lines that are parallel to the same straight line, and not in the same plane with it, are parallel to one another; therefore AD is parallel to CF; and it is equal * to it ; and AC, DF join them towards the same parts; and therefore AC is equal and parallel to DF. And because AB, BC are equal to DE, EF, each to each, and the base AC to the base DF, the angle ABC is equal to the angle DEF. Therefore, if two straight lines, &c. Q. E. D. PROPOSITION XI. * PROBLEM. To draw a straight line perpendicular to a plane, from a given point above it. Let A be the given point above the plane BH; it is required. to draw from the point A, a straight line perpendicular to the plane BH, |