Circle Segment Congruence 6A7084

1. **State the problem:** Given two congruent circles $\odot B \cong \odot Y$ and segments $AC \cong XZ$, prove that $AC \cong XZ$. 2. **Fill in the proof steps:** Statements Reasons 1. $\odot B \cong \odot Y$; $AC \cong XZ$ 1. Given. 2. Draw radii $BA$, $BC$, $YX$, and $YZ$ 2. Definition of radii in congruent circles. 3. $\triangle BAC \cong \triangle YXZ$ 3. By SSS Postulate: $BA \cong YX$, $BC \cong YZ$, and $AC \cong XZ$. 4. $\therefore AC \cong XZ$ 4. Corresponding parts of congruent triangles are congruent (CPCTC). 3. **Theorems or postulates for the statements:** 3. If $AB \cong DC$, then $AB \cong DC$. Reflexive Property of Congruence. 4. If $AB \cong DC$, then $AB \cong DC$. Reflexive Property of Congruence. 5. If $AB \cong DC$, then $\angle AOB \cong \angle DOC$. Radii of congruent circles are congruent. 6. If $\angle AOB \cong \angle DOC$, then $AB \cong DC$. Corresponding chords subtending congruent arcs are congruent. This completes the proof that $AC \cong XZ$ using the SSS Postulate and properties of congruent circles.