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

1. Z is the midpoint of XW, so by definition, XZ is congruent to ZW. 2. We are told that XZ is congruent to YZ, so YZ is congruent to ZW. 3. Thus, both XYZ and YZW are isosceles, meaning e = a and f = d. 4. Thus, we obtain that: $2e + b = 180$ And $2f + 180 - b = 180$ Adding these gives: $2(e+f) = 180 \\ e+ f = 90$ Thus, since XYZ equals e plus f, XYZ is 90 degrees.
First step : proving that $\triangle XZY and \triangle WZY$ are isosceles. •Given that $\overline{XZ}=\overline{YZ}$ and Z is the midpoint of $\overline{XW}$. •using the transitive property we conclude that $\overline{XZ}=\overline{YZ}=\overline{ZW}$ •Any triangle that has exactly two congruent sides is isosceles then $\triangle XZY and \triangle WZY$ are isosceles triangles.$\square$ Step two: •knowing that the base angles of an isosceles triangle are congruent then $\angle e=\angle a And \angle f=\angle d$. • angle Z is a straight angle then $\angle Z=\angle b+\angle c=180^{\circ}$ And $\triangle XZY=\angle e+\angle a+\angle b= 180^{\circ}$ and $\triangle WZY=\angle d+\angle f+\angle c= 180^{\circ}$ • by substitution and since $\angle e=\angle aAnd \angle f=\angle d$ Then $2\angle e+ \angle b=180^{\circ}And 2\angle f + \angle c=180^{\circ}$ we conclude that $\angle b=180-2\angle e And \angle c=180-2\angle f$ •last step by substituting $\angle b and \angle c$ in the straight angle Z equation we conclude that $180-2\angle e + 180 -2 \angle f=180^{\circ}$ $2( \angle e+ \angle f)=180^{\circ}$ $\angle e+ \angle f=90^{\circ}$ Therefore, $\angle XYW=\angle e+\angle t=90^{\circ}$ $\square$