Question

1. z is the midpoint of \\( \overline { v x } \\) and \\( \overline { w y } \\). complete the proof that \\( \triangle v y z \cong \triangle x w z \\).

statement	reason
2	\\( z \\) is the midpoint of \\( \overline { w y } \\)
3	\\( \overline { v z } \cong \overline { x z } \\)
4	\\( \overline { w z } \cong \overline { y z } \\)
5	\\( \angle v z y \cong \angle w z x \\)
6	\\( \triangle v y z \cong \triangle x w z \\)

Explanation:

Step1: Midpoint definition

Since \(Z\) is the midpoint of \(\overline{VX}\), by the definition of a midpoint, \(VZ = XZ\).

Step2: Midpoint definition

Since \(Z\) is the midpoint of \(\overline{WY}\), by the definition of a midpoint, \(WZ = YZ\).

Step3: Vertical angles

\(\angle VZY\) and \(\angle WZX\) are vertical angles. By the vertical angles theorem, \(\angle VZY\cong\angle WZX\).

Step4: SAS congruence

In \(\triangle VYZ\) and \(\triangle XWZ\), we have \(VZ = XZ\) (from step 1), \(WZ = YZ\) (from step 2), and \(\angle VZY\cong\angle WZX\) (from step 3). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle VYZ\cong\triangle XWZ\).

Answer:

1. Given
2. Given
3. Definition of midpoint
4. Definition of midpoint
5. Vertical angles theorem
6. SAS (Side - Angle - Side) congruence criterion