Proposition XXVIII. Theorem.
|(115)||If a line (E F) intersect two right lines (A B and C D), and make the external angle equal to the internal and opposite angle on the same side of the line (E G A to G H C, and E G B to G H D); or make the internal angles at the same side (A G H and C H G or B G H and D H G) equal together to two right angles, the two right lines are parallel to one another.|
First, let the angles E G A and G H C be equal; and since the angle E G A is equal to B G H (XV), the angles G H C and B G H are equal; but they are the alternate angles, therefore the right lines A B and C D are parallel (XXVII).
In the same manner the proposition can be demonstrated, if the angles E G B and G H D were given equal.
Next, let the angles A G H and C H G taken together be equal to two right angles; since the angles G H D and G H C taken together are also equal to two right angles (XIII). the angles A G H and C H G taken together are equal to the angles G H D and C H G taken together; take away the common angle C H G and the remaining angle A G H is equal to G H D; but they are the alternate angles, and therefore the right lines A B and C D are parallel (XXVII). In the same manner the proposition can be demonstrated, if the angles B G H and D H G were given equal to two right angles.
By this proposition it appears, that if the line G B makes the angle B G H equal to the supplement of G H D (84), the line G B will be parallel to H D. In the twelfth axiom (54) it is assumed, that if a line make an angle with G H less than the supplement of G H D, that line will not be parallel to H D, and will therefore meet it, if produced. The principle, therefore, which is really assumed is, that two right lines which intersect each other cannot be both parallel to the same right line, a principle which seems to be nearly self-evident.
If it be granted that the two right lines which make with the third, G H, angles less than two right angles be not parallel, it is plain that they must meet on that side of G H on which the angles are less than two right angles; for the line passing through G, which makes a less angle than B G H, with G H on the side B D, will make a greater angle than A G H with G H on the side A C; and therefore that part of the line which lies on the side A C will lie above A G, and therefore can never meet H C.
Various attempts have been made to supercede the necessity of assuming the twelfth axiom; but all that we have ever seen are attended with still greater objections. Neither does it seem ot us, that the principle which is really assumed as explained above can reasonably be objected against. See Appendix, II.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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