The problem asks whether line segment $\overline{AB}$ is congruent to line segment $\overline{XY}$ in the given circle. We need to explain our reasoning. The available options are to fill in the blanks: "Is $\overline{AB} \cong \overline{XY}$? Explain. Select Choice all Select Choice of the same circle Select Choice congruent."

1. Problem Description

The problem asks whether line segment $\overline{AB}$ is congruent to line segment $\overline{XY}$ in the given circle. We need to explain our reasoning. The available options are to fill in the blanks:

"Is $\overline{AB} \cong \overline{XY}$? Explain.

Select Choice all Select Choice of the same circle Select Choice congruent."

2. Solution Steps

From the given diagram, we can observe the following:

- $AR = XR = BR = YR$ since they are all radii of the same circle.

- $\angle ARB$ and $\angle XRY$ are vertical angles, so $\angle ARB \cong \angle XRY$.

Using the Side-Angle-Side (SAS) congruence theorem, if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent.

In triangles $\triangle ARB$ and $\triangle XRY$:

- $AR \cong XR$ (both are radii of the same circle)

- $\angle ARB \cong \angle XRY$ (vertical angles)

- $BR \cong YR$ (both are radii of the same circle)

Therefore, $\triangle ARB \cong \triangle XRY$ by SAS.

Since $\triangle ARB \cong \triangle XRY$, corresponding parts of congruent triangles are congruent (CPCTC). Hence, $\overline{AB} \cong \overline{XY}$.

The complete sentence should be:

"Is $\overline{AB} \cong \overline{XY}$? Explain.

all radii of the same circle and vertical angles are congruent."

3. Final Answer

all radii of the same circle and vertical angles are congruent.