PARALLEL. 



site angles. 2. (Prop. 19.) The greater 

 angle of every triangle has the greater 

 side opposite to it. 3. (Prop. 4.) If two 

 triangles have two sides of the one respec- 

 tive!) equal to two sides of the other, and 

 have the included angles equal, the other 

 angles will be respectively equal, viz. 

 those to which the equal sides are oppo- 

 site. 4. (Prop. 15.) If two straight lines 

 cut each other, the vertical or opposite 

 angles will be equal. 5. (Prop. 13.) If 

 a straight line meet another, the sum of 

 the adjacent angles is equal to the sum of 

 two right angles. 



6. Definition. Parallel straight lines 

 are those whose least distances from each 

 other are every where equal, 



7. Theorem 1. The perpendicular 

 drawn to a straight line from any point, 

 is the least line that can be drawn from 

 that point to the given line. 



Let C D, (Plate XII Miscell. fig. 2) be 

 a straight line drawn from C perpendicu- 

 lar to A B ; and let C E be any other 

 straight line from C to A B ; then is C D 

 less than C E. For the angle C D E 

 equals angle C D A by construction ; and 

 C DA is greater than C E D (1) ; therefore 

 C D E is greater than C E D. Hence (2) 

 C D is less than C B. 



8. Cor. 1. Hence the perpendicular 

 from any point to a straight line is the 

 true measure of the least distance of that 

 point from that line. 



9. Cor. 2. Hence (6) the perpendicu- 

 lars to one of two parallel straight lines, 

 from any points in the other, are every 

 where equal to each other. 



10. Cor. 3. Hence two parallel straight 

 lines, however far they may be produced, 

 can never meet. 



11. Theorem II. If a line meeting two 

 parallel straight lines be perpendicular to 

 one of them, it is also perpendicular to the 

 other. 



If A B, (fig. 3) be parallel to C D, and 

 E F meet them so as to be perpendicular 

 to AB, it will also be perpendicular to C D. 

 If not, draw E G perpendicular to C D, and 

 from G draw G H perpendicular to A B. 

 Then since E F and G H are both perpen- 

 dicular to A B, and are drawn from F and 

 G points in C D, G H equals E F (9). 

 Again, since angle GHB or GHE is 

 greater than angle GEH(l)EGis great- 

 er than GH (2) Hence E G is greater 

 than E F. Therefore E G is not perpen- 

 dicular to C D (7) ; and in the same man- 

 ner it may be shown, that no other line 

 can be drawn from the point E perpen- 

 dicular to CD without coinciding with 



E F. Therefore E F is perpendicular to 

 CD. 



12. Theorem HI. If two straight lines 

 be perpendicular to the same straight line, 

 they are parallel to each other. 



If A B, (fig. 4) and C I) be both perpen- 

 dicular to E F, then A B is parallel to 

 C D. If A B be not parallel to C D, let 

 G H, passing through the point E, be pa- 

 rallel to C 1). Then since E F is perpen- 

 dicular to C D, it is also perpendicular to 

 G H (11). Hence angle H E F is a right 

 angle, and therefore equal to ajigle B bl F, 

 the less to the greater, which is> absurd. 

 Therefore G H is not parallel to C I); 

 and in the same manner it may be shown 

 that no other line passing through E, 

 and not coinciding with A B, is parallel 

 to C D. Therefore A B is parallel to 

 CD 



13. Cor. Hence it appeal s, that through 

 the same point no more than one line can 

 I* drawn paraltel to the same straight 

 line. 



It may be thought necessary to remark", 

 that the preceding theorem pre-supposes 

 the admission of a postulate, that through 

 any point, not in a given straight line, a 

 straight line may be drawn parallel to 

 that straight line, or that straight line pro- 

 duced. 



14. Theorem IV. If a straight line 

 fall upon two parallel straight lines, it 

 makes the alternate angles equal to one 

 another; and the exterior angle equal to 

 the interior and opposite angle on the 

 same side ; and likewise, the two interior 

 angles upon the same side, together, 

 equal to two right angles. 



If A B, (fig, 5.) be parallel to C D, and 

 E F cut them in the points H G, then the 

 angle A H G equals the alternate angle 

 H G D ; the exterior angle E H B equals 

 the interior and opposite angle on the 

 same side, H G D ; and vhe two interior 

 angles on the same side, B N G ; and 

 H G D are together equal to two right an- 

 gles. From H draw H K perpendicular 

 to C D, and from G draw G I perpendicu- 

 lar to A B. Then since H K is perpen- 

 dicular to C D, it is also perpendicular to 

 AB (11) ; consequently G I is parallel to 

 H K (12> But H I and G K are perpen- 

 diculars to GI, from H and K, points m 

 H K; therefore (9) H I equals G K. Hence 

 in triangles G I H, H G K, the side H I 

 equals the side G K, G I equals H K (9) 

 and the included angle G I H equals the 

 included angle H K G ; therefore angle 

 I H G equals angle H G K (3). Again, 

 angle E H B equals AH G (4) ; therefore it 



