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 = ZH KD. We are to prove AB to C D. then Through the point H draw M N to CD; But .. A B, which coincides with MN, is to C D. § 68 Hyp. Ax. 1. Cons. Q. E. D. 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, EHB and CKF, and also A HE and D KF, 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 II and K, and let the EH B 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 CD. 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+ ZHKD = two rt. . 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 C D in the points H and K, and let the BHK + Z HKD equal two right angles. Then But We are to prove AB to CD. Through the point I draw MN | to CD. LNHK+ HKD = 2 rt. 4, Z (being two interior & on the same side of the secant line). 2 BHK + Z H K D = 2 rt. . $73 Hyp. Ax. 1. :. LNH K+ 2 H K D = 2 BHK + Z HKD. But .. the lines A B and M N coincide. MN is to CD ; .. A B, which coincides with MN, is to C D. Cons. Q. E D. |