Provide a step-by-step proof for the congruence of triangles ABO and CDO.

Proof of Congruence

We need to prove that ╨ ABO ≅ ╨ CDO.

1. Given information:
   • ∠A = 90°
   • ∠C = 90°
   • AO = CO (indicated by single tick marks)
   • BO = DO (indicated by double tick marks)
2. Identify congruence criteria: We have two pairs of equal sides (AO=CO, BO=DO) and one pair of equal angles (∠A = ∠C). However, these are not the included angles between the equal sides. We need to look for other angle relationships.
3. Vertical angles: Angles ∠AOB and ∠COD are vertical angles formed by the intersection of lines AD and BC. Vertical angles are always equal. Therefore, ∠AOB = ∠COD.
4. Congruence statement: We now have:
   • Side AO = Side CO
   • Angle ∠AOB = Angle ∠COD (Vertical angles)
   • Side BO = Side DO
5. Applying the SAS (Side-Angle-Side) congruence postulate: Since we have two pairs of corresponding sides equal and the included angle between them is also equal, the triangles ╨ ABO and ╨ CDO are congruent by the SAS postulate.

Conclusion: Therefore, ╨ ABO ≅ ╨ CDO.