So also, CF-DF Then in As ADF, BCF, ·.· AD=BC, and AF=BF, and DF=CF .. L DAF = 4 CBF. Again, in As AFG, BFH, I. c. p. 18. · AF=BF, and AG=BH, and ▲ FAG FBH, .. FG=FH. Then in AS FEG, FEH, L ... GE=HE, and EF is common, and FG=FH, ..L FEGL FEH. .. EF is 1 to GH. I. 4. I. c. In like manner it may be shown that EF is 1 to every st. line which meets it in the plane passing through AB, CD. .. EF is 1 to the plane, in which AB, CD are. XI. Def. 2. Q. E. D. PROPOSITION V. THEOREM. If three straight lines meet all at one point, and a straight line stand at right angles to each of them at that point, the three straight lines must be in one and the same plane. D E Let the st. line AB be to each of the st. lines BC, BD, BE, at B, the pt. where they meet. Then must BC, BD, BE be in one and the same plane. If not, let BD, BE be in one plane, and BC without it, and let a plane, passing through AB, BC, cut the plane, in which BD and BE are, in the st. line BF. Then AB, BC, BF are all in one plane. And AB is 1 to BD and BE, .. AB is XI. 2. to the plane in which BD and BE are, XI. 4. and .. AB is to BF, a st. line in that plane. XI. Def. 2. the less-the greater, which is impossible. .. BC is not without the plane, in which BD, BE are, and .. BC, BD, BE are in one and the same plane. Hyp. Q. E. D. If two straight lines be at right angles to the same plane, they must be parallel to one another. Let the st. lines AB, CD be to the same plane. Let AB, CD meet the plane in the pts. B, D. Make DE=AB, and join BE, AE, AD. Then AB is to the plane, .. AB is 1 to BD and BE, and ... each of the ¿s ABD, ABE is a rt. 2 . So also, each of the 4 s CDB, CDE is a rt. 4 . XI. Def. 2. · AB= ED, and BD is common, and rt. ▲ ABD=rt. ▲ EDB. .. DA=BE. Again, in As ABE, EDA, ·.· AB= ED, and BE DA, and AE is common, I. 4. I. c. Thus ED is to BD, AD, CD, at the pt. where they meet, and ... BD, AD, CD are all in one plane. But AB is in the plane, in which BD and AD are; and .. AB, BD, CD are all in one plane. each of the 4s ABD, CDB is a rt. 4, Then .. AB is || to CD. XI. 5. XI. 1. I. 28. Q. E. D. PROPOSITION VII. THEOREM. 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 and CD be parallel straight lines. Take any pts. E, F in AB and CD. Then must the st. line joining E and F be in the same plane as AB, CD. If not, let it be without the plane, as EGF. In the plane ABCD, in which the parallels are, draw the st. line EHF from E to F. Then the two st. lines EGF, EHF enclose a space, which is impossible. I. Post. 5. .. the st. line joining E and F cannot be out of the plane, in which the parallels AB, CD are. .. it is in that plane. Q. E. D. Note. We have proved this Proposition as Cor. IV. to Prop. I. PROPOSITION VIII. THEOREM. If two straight lines be parallel, and one of them be at right angles to a plane, the other must be at right angles to the same plane. Let AB, CD be two || st. lines, and let one of them, AB, be to a plane. Then must CD be to the same plane. Let AB, CD meet the plane in the pts. B, D; and join BD; then AB, BD, CD are all in one plane. In the plane, to which AB is 1, draw DEL to BD, .. each of the 4 s ABD, ABE is a rt. 4; Then in the AS ABD, EDB, XI. 7. XI. Def. 2. I. 29. ::: AB=ED, and BD is common, and rt. ▲ ABD=rt. ▲ EDB. .. AD=EB. Then in As ABE, EDA, ·.· AB=ED, and AE is common, and EB=AD. I. 4. I. c. Hence ED is 1 to DA, and it is also 1 to BD, by constr., .. ED is to the plane in which DA, BD are, XI. 4. and .. ED is 1 to DC, which is in that plane. XI. Def. 2. |