PROPOSITION XII. THEOREM. 69. CONVERSELY: When two straight lines are cut by a third straight line, if the alternate-interior angles be equal, the two straight lines are parallel. Let E F cut the straight lines A B and C D in the points H and K, and let the AHK = 2 HKD. then But But Through the point H draw MN | to CD; .. A B, which coincides with MN, is to C D. § 68 Hyp. Ax. 1. Cons. Q. E. D. PROPOSITION XIII. THEOREM. 70. If two parallel lines be cut by a third straight line, the exterior-interior angles are equal. Let A B and C D be two parallel lines cut by the straight line E F, in the points H and K. 71. COROLLARY. The alternate-exterior angles, EIIB and C KF, and also A HE and DK F, are equal. PROPOSITION XIV. THEOREM. 72. CONVERSELY: When two straight lines are cut by a third straight line, if the exterior-interior angles be equal, these two straight lines are parallel. Let EF cut the straight lines A B and C D in the points H and K, and let the EHB ZHKD. We are to prove AB to CD. = Through the point H draw the straight line MN || to CD. .. A B, which coincides with MN, is to C D. § 70 Hyp. Ax. 1. Cons. Q. E. D. PROPOSITION XV. THEOREM. 73. If two parallel lines be cut by a third straight line, the sum of the two interior angles on the same side of the secant line is equal to two right angles. Let A B and C D be two parallel lines cut by the straight line EF in the points H and K. ZBHK+ Z HKD =two rt. . then We are to prove PROPOSITION XVI. THEOREM. 74. CONVERSELY: When two straight lines are cut by a third straight line, if the two interior angles on the same side of the secant line be together equal to two right angles, then the two straight lines are parallel. Let EF cut the straight lines A B and CD in the points H and K, and let the BHK + Z H K D equal two right angles. Then But We Te are to prove AB to CD. Through the point H draw MN to CD. (being two interior on the same side of the secant line). § 73 Нур. :. LNH K+ZHKD=ZBHK+ZH KD. Ax. 1. Take away from each of these equals the common ▲ II K D, .. A B, which coincides with MN, is to C D. Cons. Q. E D. |