AMSCO'S Geometry. New York - Rye High School

Parallel Lines in the Coordinate Plane 343If two lines are perpendicular, the slope of one is the negative reciprocalof the slope of the other. Therefore, the slope of k is 2m1 . It is given that l 1 l 2.Then, k is perpendicular to l 2because if a line is perpendicular to one of twoparallel lines, then it is perpendicular to the other. The slope of l 2is thenegative reciprocal of the slope of k. The negative reciprocal of 2m1 is m.Therefore, the slope of l 1is equal to the slope of l 2.Is the converse of this statement true? We will again use the fact that twolines are perpendicular if and only if the slope of one is the negative reciprocalof the slope of the other to prove that it is.Theorem 9.10bIf the slopes of two non-vertical lines in the coordinate plane are equal, thenthe lines are parallel.GivenLines l 1and l 2with slope myProvel 1 l 2l 1ProofChoose any point on l 1. Through agiven point, one and only one line canbe drawn perpendicular to a givenline. Through that point, draw k, a linek l 2perpendicular to l 1. The slope of k is2m1 since two non-vertical lines areperpendicular if and only if the slopeof one is the negative reciprocal of theslope of the other. But this means thatOxl 2⊥ k because the slope of l 2is also the negative reciprocal of the slope of k.Therefore, l 1 l 2because two lines perpendicular to the same line areparallel.We can write the statements that we have proved as a biconditional:Theorem 9.10Two non-vertical lines in the coordinate plane are parallel if and only if theyhave the same slope.EXAMPLE 1The vertices of quadrilateral ABCD are A(2, 4), B(6, 2), C(2, 6), andD(1, 2).a. Show that two sides of the quadrilateral are parallel.b. Show that the quadrilateral has two right angles.