To complete the proof that $\triangle VYZ \cong \triangle WXZ$, let's analyze the provided information:

Given Information:

1. $VZ \cong WZ$ (Given)
2. $YZ \cong XZ$ (Given)
3. $\angle VZY \cong \angle WZX$ (Vertical Angle Theorem)

Reasoning:

The three statements above give us two pairs of congruent sides and a pair of congruent angles between those sides. According to the Side-Angle-Side (SAS) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Conclusion:

In step 4, we can state:

• Statement: $\triangle VYZ \cong \triangle WXZ$
• Reason: SAS Congruence Postulate

So the completed proof is as follows:

Would you like further details on any part of this proof?

Additional Questions for Deeper Understanding:

1. Why is the Side-Angle-Side (SAS) postulate sufficient for proving triangle congruence?
2. How does the Vertical Angle Theorem help in this proof?
3. What other congruence postulates or theorems could potentially be used here?
4. If one pair of sides were not congruent, could the triangles still be congruent? Why or why not?
5. How would this proof change if there was an extra pair of congruent angles instead?

Tip: Remember that vertical angles are always congruent because they are opposite each other when two lines intersect. This is a common reason used in triangle congruence proofs.