Question

part 3: choose a proof method.

1. use a paragraph, flow chart, or two - column proof to prove that \\( \overline { z x } \\) is the perpendicular bisector of side \\( \overline { w y } \\).

given: \\( \angle w z x \cong \angle y z x ; \overline { z w } \cong \overline { z y } \\)
prove: \\( \overline { z x } \\) is a perpendicular bisector of \\( \overline { w y } \\).

Explanation:

Step1: Prove $\triangle WZX\cong\triangle YZX$

Given $\angle WZX\cong\angle YZX$, $\overline{ZW}\cong\overline{ZY}$, and $\overline{ZX}\cong\overline{ZX}$ (common side). By the Side - Angle - Side (SAS) congruence criterion, $\triangle WZX\cong\triangle YZX$.

Step2: Find corresponding parts

Since $\triangle WZX\cong\triangle YZX$, then $\overline{WX}\cong\overline{YX}$ (corresponding parts of congruent triangles are congruent), so $X$ is the mid - point of $\overline{WY}$.

Step3: Find angle measure

Also, $\angle ZXW\cong\angle ZXY$. And $\angle ZXW+\angle ZXY = 180^{\circ}$ (linear pair). So $\angle ZXW=\angle ZXY = 90^{\circ}$ (if two congruent angles add up to $180^{\circ}$, each is $90^{\circ}$).

Answer:

Since $X$ is the mid - point of $\overline{WY}$ ($\overline{WX}\cong\overline{YX}$) and $\angle ZXW=\angle ZXY = 90^{\circ}$, $\overline{ZX}$ is the perpendicular bisector of $\overline{WY}$.