1. **State the problem:** Given that segment $XM$ bisects segment $AO$ and angle $X$ equals angle $M$, prove that triangles $AXI$ and $OMI$ are congruent. 2. **Identify given information and what to prove:** - $XM$ bisects $AO$ means $AX = MO$. - $\angle X = \angle M$ (given). 3. **Use the Side-Angle-Side (SAS) congruence criterion:** - We have $AX = MO$ (since $XM$ bisects $AO$). - $\angle X = \angle M$ (given). - Segment $XI = MI$ because $X$ and $M$ are points on $XM$ and $I$ is common to both triangles. 4. **Write the congruence statement:** - Triangles $AXI$ and $OMI$ have two sides and the included angle equal: $AX = MO$, $\angle X = \angle M$, and $XI = MI$. - Therefore, by SAS, $\triangle AXI \cong \triangle OMI$. 5. **Conclusion:** - Triangles $AXI$ and $OMI$ are congruent as required. This completes the proof.