Question

complete the proof that \\( \triangle vyz \cong \triangle wxz \\).

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

Explanation:

Step1: Reflexive property of congruence

A segment is congruent to itself. So, $\overline{VZ}\cong\overline{WZ}$ by reflexive property.

Step2: Given

From the figure, the markings show that $\overline{XZ}\cong\overline{YZ}$.

Step3: Vertical angles are congruent

$\angle VZY$ and $\angle WZX$ are vertical angles. So, $\angle VZY\cong\angle WZX$ (vertical angles theorem).

Step4: SAS (Side - Angle - Side) congruence criterion

In $\triangle VYZ$ and $\triangle WXZ$, we have $\overline{YZ}\cong\overline{XZ}$, $\angle VZY\cong\angle WZX$, $\overline{VZ}\cong\overline{WZ}$. So, $\triangle VYZ\cong\triangle WXZ$ by SAS.

Answer:

1. Reflexive property of congruence; 2. Given; 3. Vertical angles theorem; 4. SAS (Side - Angle - Side) congruence criterion.