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 Rare both perpen dicular to A 13, and are drawn from F and G points in C D. G H equals E F (9). Again, since angle G 1-1 B or G HE is greater than angle G E 11 (1) E G is great er than G 11 (2). Hence E G is greater than E F. Therefore E G is not perpen dicular to C I) (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 C D without coinciding with E F. Therefore E F is perpendicular to C D.
12. Theorem III. If two straight lines be perpendicular to the same straight line, they are parallel to each other.
If A B, (fig. 4) and C Ube both perpen dicular to E F, then A B is parallel to C D. If A 13 be not parallel to C D, let G passing through the point E, be rallel to C 1). Then since E. F is dicular to C D, it is also perpendicular to 11 (11). Hence angle II E F is a right angle, and therefore equal to angle B E F, the less to the greater, vi hich is absurd. Therefore G II is not parallel to C 1) ; and in the same manner it may be shown that no other line passing through E, and not coinciding with A 13, is parallel to C D. Therefore A B is parallel to CD 13. Cor. Hence it appears, that through the same point no more than one line can be drawn parallel 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 11 G equals the alternate angle 11 G D; the exterior angle E H B equals the interior and opposite angle on the same side, H G D ; and the 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 11. Then since 11 K is perpen dicular to C 1), it is also perpendicular to A B (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 in H K; therefore (9) H I equals G K. Hence in triangles G I H, H G K, the side II I equals the side G K, G I equals H K (9) and the included angle G 1 H equals the included angle H K G ; therefore angle H G equals angle 11 G K (3). Again, angle E I,1 B equals A H G (4) ; therefore it equals H G D. Lastly, B N C and IT GT) are together equal to A H G and B H G together ; and therefore (5) are equal to gether to the sum of two right angles.
15 Theorem V. If a straight line fall ing upon two other straight lines makes the alternate angles equal to one another, those two straight lines will be parallel.
Let the straight line E F, (fig. 6) which falls upon the two straight lines A C make the alternate angles A E F, E F D equal to one another, then A B is parallel to C ll. If not, through E draw G li parallel to C I). Then the alternate angle G E F equals the alternate angle E F D. But A E 1' equals E F D; therefore A E F is equal to G E F, the less to the greater. Hence G H is not parallel to C D; and in like manner it may be shown that no other line passing through the point E, and not coinciding with A B, is parallel to C I). Therefore A B is parallel to C D.
16. Cor. If a straight line, falling upon two other straight lines, makes the exte rior angle equal to the interior and oppo site one on the same side of the line ; or makes the interior angles on the same side equal to two right angles; the two straight lines shall be parallel to one another.