BOOK xi. PROP, iv. tx.] MATHEMATICS. PLANE GEOMETRY. 



593 



PROPOSITION IV. THEOREM. 

 If a, straight line (EF) be at right angles to each of tw 

 straight lines (A B, C D) at their point of intersection (E] 

 it shall also be at right angles to the plane (A D B C), whifl 

 passes through them, i.e., to the plane in which they are. 

 Take the straight lines E A, E B, E C, E D equal t< 

 each other. Join A D am 

 BC. In the plane AD BC 

 draw through E any l 

 G K H, meeting A D and 

 B C in G and H ; we are 

 to show that F E is perpen 

 dicular to G H. For join 

 FA, FB, FC, FD, FG 

 F n. Now, because in the triangles A E 1), B E C, the 

 sides A E, E D = the sides C E. E B, and the angle A ED 



is i. = angle C E B,* .'. A D B C, and angle DAE 

 + 4j. = angle EBC.f Again, in triangles AGE, 

 .151. E B H we have the angle A E G = anyle B EH ;* 



and by what we have proved, angle G A E = E B H, and 



the side A E = side E B ; .-. G E = E H, and A G = 



+ 261. B H.f Again, since the sides A E, EFare 



equal to the sides D E, E F each to each, and the right 



AX. 11 1. angle A E F = the right angle DBF;* 

 + 41. A F = D F.f Similarly A F = F B, and E 



F C. Hence in the triangles D A F, B C F we have the 

 si'les DA, A F = to the sides C B, B F, each to each, 

 and the base D F = the base F C ; .'. the angle D A F = 



si. the angle FBC.* Again, in the triangles 

 G A F, H H F we have (by what we have already proved) 

 the sides G A, A F = the sides H B, B F, each to each, 



a an-le G A F = angle H B F ; .'.base G F = base 



+ 4i. FH.t 



Hence (by what we have now proved), in the triangles 

 G E F, H E F we have the sides G K, E F = the sides 

 H E, E F, each to each, and the base F G = the base 



8 i. F H, .'. the angle G EF = the angle H E F,* 

 + 8 Def. I. which are therefore each right angles, t The 



saruu proof applius to any other line drawn through E, 

 in the plane A B C. Hence K F is perpendicular to the 



1 D,t. XI. plane.* Q. E. D.' 



PROPOSITION V. THEOKKM. 



// three straight lines (B C, B D, B F) meet all in one 

 paint (B), and a straight line (B A) staivls at right 

 angles to each of them at that point, these three straight 

 lines are in one and the same plane. 

 For if not, suppose the plane which passes through 

 B F and B D not to pans 

 through B C, and suppose the 

 plane passing through A B 

 and BC to cut the former 

 plane in B E, then the straight 

 lines BF, B D, B E are in the 

 same plane, viz., the one 

 passing through B D, B F, 

 and A B is at right angles to 

 B D, BE, and .'.is also at 

 -I XI. right angles to B E. * 

 Now, the angle A B C is a 

 + 11 Ax. I. right angle, .'.angle A B C = angle A B F,t 

 and they are both in the same plane, which is impos- 



9 AX. i. sible.*.'.B C, B D, B Fmust beinthegame 

 plane, Q. E. D. 



PROPOSITION VI. THEOREM. 



If two straight lines (A H, C D) are at ri-jht angles to the 

 "ie plane (BDE), they shall be parallel to one another. 

 B, D are the points in which the lines meet the plane. 

 Join B D, and at D draw D E 

 in the plane perpendicular to 

 BD. MakeDE = AB. Join 

 AD, BE, AE. Now, since 

 A B is perpendicular to plane, 

 A B D and ABE are right 

 angles, then in triangles A B D, 

 BDE the sides A B, B D = 

 sides DE, D B, each to each, 

 and angle A B D=angle BDE, 

 each being a right angle, .'.BE 



a 



41. = A D.* Then in triangles A D E, A B E, the 

 sides AD, D E = the sides E B, B A, each to each, and 

 the base A E common, .'. angle A B E = angle A D E.f 



+ 81. But angle A B E is a right angle, .'. A D E is a 

 right angle. Now, because C D is perpendicular to the 



3 Def. XI. plane, C D E is a right angle ; * so that 

 E D is at right angles to the three lines B D, AD, CD, 



+ 5 XI. which are therefore in the same plane ;t but 

 the plane which contains A D, D B contains AB,* .'. 



2 xi. A B, B D, D C are in the same plane ; now, 

 angles ABD and BDC are right angles, and are .'. 

 together equal to two right angles, .'. A B is parallel to 



+ 281. CD.f Q.E. D. 



PROPOSITION VII THEOREM. 



If two straight lines (A B, C D) are parallel, the straight 

 line drawn from any point (E) in the one, to any point 

 (F) in the other, is in the same plane with the parallels. 

 For if not, suppose E G F to be the straight line join- 

 ing them, and suppose it do 

 r a not fall in the plane. Since 



E and F are points in the 

 H\\Q plane, we can join them by a 



straight line, which lies wholly 

 O in the plane. Let this line be 

 EHF. ThenEHFandEGF 

 are two straight lines, inclosing a space, which is impos- 

 sible. Q. E.D. 



PROPOSITION VIII. THEOREM. 

 If two straight lines (A B, C D) are parallel, and one of 

 them (A B) is at right angles to a given plane (H D E), 

 the other ahull also be at right angles to the satne plane. 

 Let the lines meet the plane in B and D. Join B D. 

 Then A B, B 1), D C are in one 

 7 XI. plane. * Draw D E at 

 right angles to B D, and in the 

 + 111. plane B D E. t Take A B 

 = 1)E. Join AD, AE, BE. 

 Now, A B being perpendicular to 

 the plane, is perpendicular to HI) 

 Def. xi. and BE.* Now, BD 

 meets the parallel lines A B, CD, 

 .'.the angles ABD, BDC are 

 together equal to two right an- 

 + 291. gles.f But ABD is a 

 ri,-ht angle, .'. B D C is a right angle. 



Again, in triangles ABD, B D K, the sides A B, B D 

 = the sides E D, D B, and the right angle A B D = the 



4 I. right angle B 1) E ; .'. A D = B E. * Hence in 

 triangles A B E, A D E we have the sides A B, B E = 



he sides El), DA, and the base A E is common, .'. 



+ 81. angle A B E = angle A D E.f But A B E is a 



right angle, .'. A D E is a right angle ; .'. E D is at right 



.ugles to the lines B D and D A, and .'. is at right angles 



4 XI. to the plane passing through them,* and /. since 

 y D is in the same plane with A D, B D, angle C D E is 



+ Def. XI. a right anjjle. t But we have already seen that 

 j D C is a right angle, .'.CD is at right angles to the 



4X1. plane passing throughBD, D E,*t.e.,is atright 

 angles in the plane B D E. Q. E. D. 



PROPOSITION IX. THEOREM. 



/ two straight lines (A B, C D) are each parallel to the 

 same straight line (K F), but not both in the same plane 

 with it, they are parallel to one another. 



For, tale any point G in E F, and from G in the 



n i. plane AB, EF draw GH perpendicular to EF ; 



and likewise from G in plane 



!- ? EF, CD, draw GK parpuu- 



; j^X^ c dicular to E F. 



jl| j .X Then because G F is at right 



' angles to G H aud G K, it is at 



right angles to the plane 



+ 4 XI. H G K ;t and since G F is at right angles to 



he plane H G K, and H B is parallel to G F, .'. H B is 



8X1. at right angles to HGK.* Similarly K D is 



.t right angles to H G K, .'. H B and KD are parallel,! 



4o 



