This is a two-column proof showing that $\overline{VX} \cong \overline{VW}$. Let’s complete the steps systematically.

Completed Proof:

1. $Y$ is the midpoint of $\overline{WX}$ | Given
2. $\overline{VY} \perp \overline{WX}$ | Given
3. $\angle VYW \cong \angle VYX$ | All right angles are congruent
4. $\overline{WY} \cong \overline{XY}$ | Definition of midpoint
5. $\overline{VY} \cong \overline{VY}$ | Reflexive Property of Congruence
6. $\triangle VYW \cong \triangle VYX$ | SAS Postulate
7. $\overline{VX} \cong \overline{VW}$ | Corresponding parts of congruent triangles (CPCTC)

Explanation:

• Step 6: By the Side-Angle-Side (SAS) congruence postulate, the triangles $\triangle VYW$ and $\triangle VYX$ are congruent because $\overline{WY} \cong \overline{XY}$, $\angle VYW \cong \angle VYX$, and $\overline{VY} \cong \overline{VY}$ (reflexive).
• Step 7: Once the triangles are proven congruent, corresponding parts (like $\overline{VX}$ and $\overline{VW}$) are also congruent.

Let me know if you would like additional clarifications!

Related Questions:

1. Why is the SAS Postulate crucial for proving congruence in this problem?
2. How does the definition of midpoint support the proof?
3. Can you apply the Reflexive Property of Congruence in other geometric contexts?
4. What are some alternative ways to prove triangle congruence (e.g., ASA, SSS)?
5. How can CPCTC be applied to solve real-world geometry problems?

Tip:

Always ensure each congruence postulate (e.g., SAS, ASA, etc.) is properly justified with corresponding statements and reasons. This approach avoids missing logical steps.