## Elementary Geometry for College Students (6th Edition)

1-In a scalene triangle XYZ in which $\overline{ZW} bisects \angle XZY$ let us assume that ZW is perpendicular to segment XY. 2- segment ZW separates the $\triangle XYZ$ into two right triangles$( \triangle ZWY and \triangle ZWX )$ 3- these two triangle by our assumption they must be congruent by ASA since $\angle ZWY = \angle ZWX= 90^{\circ}, \overline{ZW}=\overline{ZW}$ by identity and $\angle YZW = \angle WZX$ by definition of angle bisector. 5- therefore $\angle Y = \angle X$ by CPCTC which contradicts the hypothesis, also $\overline{YZ}= \overline{XZ}$by CPCTC contradicts the hypothesis that triangle XYZ is a scalene triangle. 6- thus, the assumed statement must be false and segment ZW is not perpendicular to XY.