To complete the two-column proof for proving that triangles $\triangle XVW$ and $\triangle ZVW$ in the figure are congruent, let’s analyze the diagram and go through each step with the appropriate statements and reasons.

Analysis

From the diagram, we observe that:

1. $XV = ZV$ and $XW = ZW$ (indicated by tick marks).
2. $\angle XVW$ is congruent to $\angle ZVW$ (indicated by the arc marks).
3. The side $VW$ is shared between both triangles.

Given these observations, we can now complete the two-column proof.

Proof

Statements

Each selection fits as follows:

1. Statement: $XV = ZV$, $XW = ZW$, $\angle XVW \cong \angle ZVW$; Reason: Given
2. Statement: $VW = VW$; Reason: Reflexive property
3. Statement: Triangles $\triangle XVW$ and $\triangle ZVW$; Reason: Given
4. Reason: Definition of congruent triangles
5. Reason: Definition of congruent triangles

Would you like further details on this proof?

Questions to Deepen Understanding

1. How can the Side-Angle-Side (SAS) postulate be used to prove congruence in other triangle pairs?
2. Why is the reflexive property essential in this proof?
3. Can other triangle congruence postulates, like ASA or AAS, be used here? Why or why not?
4. What does the term "definition of congruent triangles" specifically imply in a proof?
5. How would we approach this proof if we were asked to use transformations?

Tip

Remember that the reflexive property (an object being equal to itself) is very useful in geometry, especially for proving congruence in triangles that share a side.