40 V. PARALLEL LINES The other two cases can be reduced to the first: 2◦. If 3 = 5, this is equivalent to saying that 3 is the supplement of 6, or that the interior angles on the same side are supplementary. 3◦. If 6 = 2, then again 3 and 6 are supplementary, because 3 is the supplement of 2. This theorem can be used to prove that two lines are parallel. Corollary. In particular, two lines perpendicular to a third line are parallel. 39. Theorem. Through any point not on a given line, a parallel to the given line can be drawn. Consider point O and line xy (Fig. 36). If we join point O to any point A of xy, then line OO , which makes an angle with OA such that AOO + OAy is equal to two right angles, will be parallel to xy. 40. Because the preceding construction can be made in infinitely many ways (since point A can be chosen anywhere on line xy), it would seem that there are infinitely many different parallels. This is not true, however, if we adopt the following axiom: Axiom. Through any point not on a given line, only one parallel to the given line can be drawn.2 Corollaries. I. Two distinct lines parallel to a third line are themselves par- allel since if the two lines had a point in common, two lines would pass through it, each parallel to the third line. II. If two lines are parallel, any third line which intersects one of them must intersect the other, otherwise two parallels to the second line would intersect each other.3 41. The most important proposition in the theory of parallels is the following converse of the theorem in 38. Converse. When two parallel lines are intersected by the same transversal: 1◦. The interior angles on the same side are supplementary 2◦. The alternate interior angles are equal 3◦. The corresponding angles are equal. The proof is the same in all three cases. Consider the parallel lines AB, CD intersected by the transversal EF x (Fig. 37). We claim, for example, that the corresponding angles xEB and xF D are equal. Indeed, we can construct, at point E, an angle xEB equal to angle xF D. Line EB will then be parallel to CD, and therefore it will coincide with EB. Corollaries. I. If two lines determine, with a common transversal, two in- terior angles on the same side which are not supplementary, then the two lines are not parallel, and they intersect on the side of the transversal where the sum of the interior angles is less than two right angles. II. When two lines are parallel, any line perpendicular to one of them is per- pendicular to the other. 2 This axiom is known as Euclid’s Postulate. In fact it should be viewed as part of the definition in fundamental notions. (See Note B at the end of this volume.) 3 We have here another example of a proof by contradiction (see 24, Remark I.)
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