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To solve this problem, we need to prove the **Congruent Supplements Theorem**, which states that:

- **If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.**

### Given:
- \( \angle HOD \cong \angle ZUR \)
  
### Goal:
- Prove \( \angle DOC \cong \angle RUJ \).

### Diagram Analysis:
From the diagram:
1. \( \angle HOD \) and \( \angle DOC \) are adjacent, sharing the common ray \( \overrightarrow{OD} \), which implies that \( \angle HOD \) and \( \angle DOC \) are supplementary (they add up to 180°).
   
2. Similarly,

Prove the Congruent Supplements Theorem: Given that ∠HOD ≅ ∠ZUR, prove that ∠DOC ≅ ∠RUJ.

This geometry problem involves proving the Congruent Supplements Theorem, which states that if two angles are supplements of congruent angles, then the two angles are congruent. The problem uses given congruent angles ∠HOD ≅ ∠ZUR and asks to prove that ∠DOC ≅ ∠RUJ.

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