Question

statements

1. \\( \overline { w y } \cong \overline { z x } \\)
2. \\( \overline { w x } \cong \overline { z y } \\)
3. \\( \overline { x y } \cong \overline { x y } \\)
4. \\( \triangle w x y \cong \triangle z y x \\)
5. \\( \angle w \cong \angle z \\)

reasons

1. \\( \frac { \text { given } } { } \\)
2. given
3. \\( \frac { \text { reflexive property } } { } \\)
4. \\( \frac { \text { sss } } { } \\)
5. \\( \frac { \text { cpctc } } { } \\)

Explanation:

Step1: Identify the given information

We are given that \(\overline{WY}\cong\overline{ZX}\) (Statement 1) and \(\overline{WX}\cong\overline{ZY}\) (Statement 2).

Step2: Use the reflexive property

For Statement 3, \(\overline{XY}\cong\overline{XY}\) by the Reflexive Property of Congruence (a segment is congruent to itself).

Step3: Apply the SSS (Side - Side - Side) congruence criterion

Since we have three pairs of congruent sides (\(\overline{WY}\cong\overline{ZX}\), \(\overline{WX}\cong\overline{ZY}\), \(\overline{XY}\cong\overline{XY}\)), by the SSS congruence criterion, \(\triangle WXY\cong\triangle ZYX\) (Statement 4).

Step4: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

After proving \(\triangle WXY\cong\triangle ZYX\), for Statement 5, \(\angle W\cong\angle Z\) because corresponding parts of congruent triangles are congruent.

Answer:

The two - column proof is a valid way to show the congruence of \(\angle W\) and \(\angle Z\) using the given side congruences, the reflexive property, SSS congruence, and CPCTC.