The problem asks to prove that line segment $\overline{GH}$ is congruent to line segment $\overline{WV}$ using the given flowchart. The diagram shows two triangles, $\triangle GIH$ and $\triangle UWV$. From the diagram, we can identify the following congruent parts: $\angle G \cong \angle W$, $\angle I \cong \angle U$, and $\overline{GI} \cong \overline{WU}$.

1. Problem Description

The problem asks to prove that line segment $\overline{GH}$ is congruent to line segment $\overline{WV}$ using the given flowchart. The diagram shows two triangles, $\triangle GIH$ and $\triangle UWV$. From the diagram, we can identify the following congruent parts: $\angle G \cong \angle W$, $\angle I \cong \angle U$, and $\overline{GI} \cong \overline{WU}$.

2. Solution Steps

We are given $\angle G \cong \angle W$, $\angle I \cong \angle U$, and $\overline{GI} \cong \overline{WU}$.

Since two angles of a triangle are congruent to two angles of another triangle, the third angles are congruent by the Third Angles Theorem. Therefore, $\angle H \cong \angle V$.

Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Since $\overline{GI} \cong \overline{WU}$, $\angle G \cong \angle W$ and $\angle I \cong \angle U$, we have $\triangle GIH \cong \triangle UWV$ by ASA (Angle-Side-Angle) Congruence Postulate.

ASA Congruence Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Since $\triangle GIH \cong \triangle UWV$, we have corresponding parts of congruent triangles are congruent (CPCTC). Therefore, $\overline{GH} \cong \overline{WV}$.

CPCTC: Corresponding parts of congruent triangles are congruent.

Here's the completed flowchart:

1. Statement: $\angle G \cong \angle W$

Reason: Given

2. Statement: $\overline{GI} \cong \overline{WU}$

Reason: Given

3. Statement: $\angle I \cong \angle U$

Reason: Given

4. Statement: $\triangle GIH \cong \triangle UWV$

Reason: ASA Congruence Postulate

5. Statement: $\overline{GH} \cong \overline{WV}$

Reason: CPCTC

3. Final Answer

$\overline{GH} \cong \overline{WV}$