PROPOSITION X. THEOREM. 106. If two parallel lines are cut by a third straight line, the exterior-interior angles are equal. Let AB and CD be two parallel lines cut by the straight line EF, in the points H and K. 107. COR. The alternate-exterior angles EHB and CKF, and also AHE and DKF, are equal. Ex. 4. If an angle is bisected, and if a line is drawn through the vertex perpendicular to the bisector, this line forms equal angles with the sides of the given angle. Ex. 5. If the bisectors of two adjacent angles are perpendicular to each other, the adjacent angles are supplementary. PROPOSITION XI. THEOREM. 108. CONVERSELY: When two straight lines are cut by a third straight line, if the exterior-interior angles are equal, these two straight lines are parallel. Let EF cut the straight lines AB and CD in the points H and K, and let the ZEHB=LHKD. Proof. Suppose MN drawn through H || to CD. § 101 .. AB, which coincides with MN, is || to CD. Cons. Q. E. D. Ex. 6. The bisector of one of two vertical angles bisects the other. Ex. 7. The bisectors of the two pairs of vertical angles formed by two intersecting lines are perpendicular to each other. PROPOSITION XII. THEOREM. 109. If two parallel lines are cut by a third straight line, the sum of the two interior angles on the same side of the transversal is equal to two right angles. Let AB and CD be two parallel lines cut by the straight line EF in the points H and K. Substitute HKD for 2 EHB in the first equality; then = 2 BHK+2 HKD 2 rt. 4. Q. E. D. Ex. 8. If the angle AHE is an angle of 135°, find the number of degrees in each of the other angles formed at the points H and K. Ex. 9. Find the angle between the bisectors of adjacent complementary angles. PROPOSITION XIII. THEOREM. 110. CONVERSELY: When two straight lines are cut by a third straight line, if the two interior angles on the same side of the transversal are together equal to two right angles, then the two straight lines are parallel. Let EF cut the straight lines AB and CD in the points H and K, and let the BHK+2HKD equal two right angles. But Proof. Suppose MN drawn through H to CD. Then Z NHK+2 HKD=2 rt. 4, (being two interior of 's on the same side of the transversal). from each of these equals the common / HKD; $ 109 Hyp. ::Z NHK+2 HKD=2 BHK+2 HKD. Ax.1 .. AB, which coincides with MN, is to CD. Cons. Q. E. D. PROPOSITION XIV. THEOREM. 111. Two straight lines which are parallel to a third straight line are parallel to each other. § 102 Since CD and EF are ||, HK is 1 to CD, (if a straight line is Since AB and ..Z HOB=/ HPD, (each being a rt. 4). .. AB is to CD, $ 108 (when two straight lines are cut by a third straight line, if the ext.-int. ▲ are equal, the two lines are parallel). Q. E. D. Ex. 10. It has been shown that if two parallels are cut by a transversal, the alternate-interior angles are equal, the exterior-interior angles are equal, the two interior angles on the same side of the transversal are supplementary. State the opposite theorems. State the converse theo rems. |