To complete the proof that $\overline{UY} \cong \overline{VW}$, we need to fill in the missing steps in the proof table.

Here’s the reasoning for each step:

1. $\overline{VW} \perp \overline{WX}$ - Given
2. $\overline{XY} \perp \overline{UY}$ - Given
3. $\overline{XY} \cong \overline{WX}$ - Given
4. $\angle UXY \cong \angle VXW$ - Given

Now, continuing with the reasoning:

5. $\overline{UY} \cong \overline{VW}$ - Since the triangles $\triangle UXY$ and $\triangle VXW$ have two pairs of congruent sides and the included angles are congruent, we can conclude that $\triangle UXY \cong \triangle VXW$ by the SAS (Side-Angle-Side) Postulate.

6. $\triangle UXY \cong \triangle VXW$ - Reason: SAS Postulate

7. $\overline{UY} \cong \overline{VW}$ - Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

This completes the proof.

Would you like further details on any step? Here are some additional questions to consider:

1. Why is the SAS Postulate used in this proof rather than other congruence postulates?
2. What is the importance of perpendicularity in establishing triangle congruence here?
3. How would this proof change if we used different angles or sides in the proof?
4. What other postulates or theorems could be relevant if additional information were given?
5. How does CPCTC assist in concluding segment congruence?

Tip: When proving triangles congruent, always check for corresponding parts and congruence postulates (SSS, SAS, ASA, AAS, HL) that apply to the given information.