AMSCO'S Geometry. New York - Rye High School

Proving Lines Parallel 3299-1 PROVING LINES PARALLELYou have already studied many situations involving intersecting lines that lie inthe same plane. When all the points or lines in a set lie in a plane, we say thatthese points or these lines are coplanar. Let us now consider situations involvingcoplanar lines that do not intersect in one point.DEFINITIONParallel lines are coplanar lines that have no points in common, or have all pointsin common and, therefore, coincide.The word “lines” in the definition means straight lines of unlimited extent.We say that segments and rays are parallel if the lines that contain them areparallel.We indicate that AB gis parallel to CD gby writing CAB g CD g. The parallel lines AB gand CD gextended indefinitelynever intersect and have no points in common.AThe parallel lines AB gand CD gmay have all points incommon, that is, be two different names for the same line.A BDBC DA line is parallel to itself. Thus, AB g AB g, CD g CD gandAB g CD g.In Chapter 4, we stated the following postulate: Two distinct lines cannot intersect in more than one point.This postulate, together with the definition of parallel lines, requires thatone of three possibilities exist for any two coplanar lines, AB gand CD g:1. AB gand CD ghave no points in common.AB gand CD gare parallel.2. AB gand CD ghave only one point in common.AB gand CD gintersect.3. AB gand CD ghave all points in common.AB gand CD gare the same line.These three possibilities can also be stated in the following postulate:Postulate 9.1Two distinct coplanar lines are either parallel or intersecting.