ELEMENTS OF GEOMETRY. SUPPLEMENT. BOOK IL OF THE INTERSECTION OF PLANES. DEFINITIONS. A I. when it makes right angles with every straight line which it meets in that plane. II. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the III. tained by that straight line, and another drawn from the point in IV. contained by two straight lines drawn from any, the same point in Z V. Two planes are said to have the same, or a like inclination to one an other, which two other planes have, when the angles of inclination above defined are equal to one another. VI. 1 A straight line is said to be parallel to a plane, when it does not meet the plane, though produced ever so far, VII. Planes are said to be parallel to one another, which do not meet, though produced ever so far. VIII. A solid angle is an angle made by the meeting of more than two plané angles, which are not in the same plane in one point. с PROP. I. THEOR. One part of a straight line cannot be in a plane and an other part about it. Ifit be possible, let AB, part of the straight line ABC be in the plano, and the part BC above it: and since the "straight line AB is in the plane, it can be produced in that plane (2. Post. 1.): let it be produced to D. Then ABC and ABD are two straight lines, and they have the common segment AB, which is impossible (Cor. def. 3. 1.).' Therefore ABC iş not a straight line. Wherefore one part, &c. Q. E. D. А B D PROP. II. THEOR. Any three straight lines which meet one another, not in the same point, are in one plane. Let the three straight lines AB, CD, CB meet one another in tlie points B, C and E; AB, CD, CB are in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced, if necessary, until it pass through the point C: Then, because the points E, C are in this plane, the straight line EC is in it (def. 5. 1.): for the same reason, the straight line BC is in the same; and, by the hypothesis, EB_is in it; therefore the three straight lines EC, CB, BE are in one plane: but the whole of the lines DC, E AB, and BC produced, are in the same plane with the parts of them EC, EB, BC (1. 2. Sup.). Therefore AB, CD, CB, are all in one plane. Wherefore, &c. 'Q. E. D. Cor. It is manifest, that any two straight lines which cut one an. other are in one plane: Also, that any three points whatever are in one plane. PROP. III. THEOR. B If two planes cut one another, their common section is a straight line. Let two planés AB, BC cut one another, and let B and D be two points in the line of their common section. From B to D draw the straight line BD; and because the points B and D are in the plane AB, the straight line BD is in that plane (def. 5. 1.): for the same reason it is in the plane CB; the straight line BD is therefore common to the planes AB and BC, or it is the common section of these planes. Therefore, &c. Q. E. D. B PROP. IV. THEOR. If a straight line stand at right angles to each of two straight lines in the point of their intersection, it will also be at right angles to the plane in which these lines are. Let the straight line AB stand at right angles to each of the straight lines EF, CD in A, the point of their intersection : AB is also at right angles to the plane passing through EF, CD. Through A draw any line AG in the plane in which are EF and CD; let G be any point in that line; draw GH parallel to AD; and make HF=HA, join FG; and when produced let it meet CA in D; join BD, BG, BF. Because GH is parallel to AD, and FH=HA; therefore FG=GD, so that the line DF is bisected in G. And because BAD, is a right angle, BD'=AB'+AD' (47.1.); and for the same reason, BF:=AB9+ AF%, therefore BD + BF = 2AB+ AD% + AF%; and because DF is bisected in G. (A. 2.), ADS + AF:=2AG”+ 2GF, therefore BD+BF = 2AB + 2AG'+ 2GF. But BD + BF»= (A. 2.) 2BGS + 2GF, therefore 2BG” + 2GFʻ = AB: + F H G 2AG: +2GF, and taking 2GF from both, 2BG'=2 AB +2AG, or BGʻ=AB'+AG”; wherefore BAG (48. 1.) is a right angle. Now AG is any straight line drawn in the plane of the lines AD, AF; and when a straight line is at right angles to any straight line which it meets with in a plane, it is at right angles to the plane itself (def. 1. 2. Sup.). AB is therefore at right angles to the plane of the lines AF, AD. Therefore, &c. Q.Ě. D. PROP, 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 are in one and the same plane. If not, let BD. and BE, if possible, 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, shall be a straight (3. 2. Sup.) 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 because AB stands at right angles to each of the straight lines BD, BE, it is also at right angles (4.2. Sup.) to the plane passing through them; and therefore makes right angles with every straight line meeting it in that plane; 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 an: gle ABF is equal to the angle ABC and they are both in the same plane, which is 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. AN F B 1 PROP. VI. THEOR. Two straight lines which are at right angles to the same plane, are parallel to one another. Let the straight lines AB, CD be at right angles to the same plane BDE; AB is parallel to CD. с B Let them meet the plane in the points B, D. Draw DE at right angles to DB, in the plane BDE, and let E be a any point in it: Join AE, AD, EB. Because ABE is a right angle, ABS + BE-= (47. 1.) AE', and because BDE is a right angle, BE = BD + DE'; therefore ABS + BD: + DE AF”; now, ABS +BD’=AD%, because ABD is a right angle, therefore AD DE:=AE, and ADE is therefore a (48. 1.) right angle. Therefore ED is perpendicular to the three lines BD, DA, DC, whence these lines are in one plane (5. 2. Sup.). But AB is in the plane in which are BD, DA, because any three straight lines, which meet one another, are in one plane (2. 2. Sup.): Therefore AB, BD, DC are in one plane; and each of the angles ABD, BDC is a right angle; therefore AB is parallel (28. 1.) to CD. Wherefore, if two straig.It lines, &c. Q. E. D. А. с G PROP. VII. THEOR. If two straight lines be parallel, and one of them at right angles to a plane; the other is also 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 is at right angles to the same plane. For, if CD be not perpendicuJar to the plane to which AB is perpendicular, let DG be perpendicujar to it. Then (6. 2. Sup.) DG is parallel to AB: DG and DC therefore are both parallel tó AB, and are draw through the same point D, which is impossible (11. Ax. 1.). Therefore, &c. Q. E. D. PROP. VIII. THEOR. Two straight lines which are each of them parallel to the same straight line, though not both 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. In EF take any point G, from which 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 ĠK at right angles to the F |