If a right line (EF) intersecting two right lines (AB, CD) makes the exterior angle (EGB) equal to its corresponding interior angle (GHD), or makes two interior angles (BGH, GHD) on the same side equal to two right angles, the two right lines are parallel. Dem.—1. Since the lines AB, EF intersect, the angle AGH is equal to EGB [xv.]; but EGB is equal to GHD (hyp.); therefore AGH is equal to GHD, and they are alternate angles. Hence [xxvii.] AB is parallel to CD. 2. Since AGH and BGH are adjacent angles, their sum is equal to two right angles [xiii.]; but the sum of BGH and GHD is two right angles (hyp.); therefore rejecting the angle BGH we have AGH equal GHD, and they are alternate angles; therefore AB is parallel to CD [xxvii.]. PROP. XXIX.—Theorem. If a right line (EF) intersect two parallel right lines (AB, CD), it makes—1. the alternate angles (AGH,GHD) equal to one another; 2. the exterior angle (EGB) equal to the corresponding interior angle (GHD); 3. the two interior angles (BGH, GHD) on the same side equal to two right angles. Dem.—If the angle AGH be not equal to GHD, one must be greater than the other. Let AGH be the greater; to each add BGH, and we have the sum of the angles AGH, BGH greater than the sum of the angles BGH, GHD; but the sum of AGH, BGH is two right angles; therefore the sum of BGH, GHD is less than two right angles, and therefore (Axiom xii.) the lines AB, CD, if produced, will meet at some finite distance: but since they are parallel (hyp.) they cannot meet at any finite distance. Hence the angle AGH is not unequal to GHD—that is, it is equal to it. 2. Since the angle EGB is equal to AGH [xv.], and GHD is equal to AGH (1), EGB is equal to GHD (Axiom i.). 3. Since AGH is equal to GHD (1), add HGB to each, and we have the sum of the angles AGH, HGB equal to the sum of the angles GHD, HGB; but the sum of the angles AGH, HGB [xiii.] is two right angles; therefore the sum of the angles BGH, GHD is two right angles. Exercises. 1. Demonstrate both parts of Prop. xxviii. without using Prop. xxvii. 2. The parts of all perpendiculars to two parallel lines intercepted between them are equal. 3. If ACD, BCD be adjacent angles, any parallel to AB will meet the bisectors of these angles in points equally distant from where it meets CD. 4. If through the middle point O of any right line terminated by two parallel right lines any other secant be drawn, the intercept on this line made by the parallels is bisected in O. 5. Two right lines passing through a point equidistant from two parallels intercept equal portions on the parallels. 6. The perimeter of the parallelogram, formed by drawing parallels to two sides of an equilateral triangle from any point in the third side, is equal to twice the side. 7. If the opposite sides of a hexagon be equal and parallel, its diagonals are concurrent.