AMSCO'S Geometry. New York - Rye High School

Properties of Parallel Lines 3359-2 PROPERTIES OF PARALLEL LINESIn the study of logic, we learned that a conditional and its converse do notalways have the same truth value. Once a conditional statement has beenproved to be true, it may be possible to prove that its converse is also true. Inthis section, we will prove converse statements of some of the theorems provedin the previous section. The proof of these converse statements requires the followingpostulate and theorem:Postulate 9.2Through a given point not on a given line, there exists one and only one lineparallel to the given line.Theorem 9.5If, in a plane, a line intersects one of two parallel lines, it intersects the other.Given AB g CD gand EF gintersects AB gEat H.Prove EF gintersects CD g.Proof Assume EF gdoes not intersect CD g. ThenFEF g CD g. Therefore, through H, a pointnot on CD g, two lines, AB gand EF gareeach parallel to CD g. This contradicts theCDpostulate that states that through a given point not on a given line, one andonly one line can be drawn parallel to a given line. Since our assumption leadsto a contradiction, the assumption must be false and its negation, EF gintersectsCD gmust be true.Now we are ready to prove the converse of Theorem 9.1a.AHBTheorem 9.1b If two parallel lines are cut by a transversal, then the alternate interior anglesformed are congruent.Given AB g CD g, transversal EF gintersects AB gat E andECD gat F.A 1 BProve1 2C2FD