Question

1 z is the midpoint of \\( \overline{wy} \\) given
2 z is the midpoint of \\( \overline{vx} \\) given
3 \\( \overline{wx} \cong \overline{vy} \\) given
4 \\( \overline{wz} \cong \overline{yz} \\) definition of midpoint
5 \\( \overline{vz} \cong \overline{xz} \\) definition of midpoint
6

Explanation:

Step1: Recall the SAS (Side - Angle - Side) Congruence Theorem

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Identify the triangles and their corresponding parts

We have \(\triangle YVZ\) and \(\triangle XWZ\).
We know that \(WZ\cong YZ\) (from statement 4), \(VZ\cong XZ\) (from statement 5), and \(\angle YZV\cong\angle XZW\) (vertical angles are congruent).

Step3: Apply the SAS Congruence Theorem

Since \(WZ\cong YZ\), \(VZ\cong XZ\), and \(\angle YZV\cong\angle XZW\), by the SAS (Side - Angle - Side) congruence theorem, \(\triangle YVZ\cong\triangle XWZ\)

Answer:

\(\triangle YVZ\cong\triangle XWZ\) (by SAS)