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This problem asks to prove the **Congruent Supplements Theorem**, which states that if two angles are supplementary to two congruent angles, then the original two angles are congruent.

We are given:
- \( \angle HOD \cong \angle ZUR \)
- We need to prove: \( \angle DOC \cong \angle RUJ \)

### Steps to solve:

1. **Restate the congruency**: From the given, \( \angle HOD \cong \angle ZUR \).
2. **Supplementary Angles**: Since \( \angle HOD \) and \( \angle DOC \) are supplementary (they form a straight line along \( \overrightarrow{OD} \)), we can say:
   \[
   \angle HOD + \angle DOC = 180^\circ
   \]
   Similarly, for the other pair, \( \angle ZUR \) and \( \angle RUJ \) are supplementary:
   \[
   \angle ZUR + \angle RUJ = 180^\circ
   \]
3. **Congruency of Supplements**: Since \( \angle HOD \cong \angle ZUR \) and both are supplementary to their respective adjacent angles, \( \angle DOC \) and \( \angle RUJ \) must be congruent as well, by the Congruent Supplements Theorem:
   \[
   \angle DOC \cong \angle RUJ
   \]

This completes the proof.

Would you like further explanation or details?

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Here are 5 related questions to explore:
1. How do we define supplementary angles?
2. Can two angles be congruent but not supplementary? Why or why not?
3. What is the definition of the Congruent Complements Theorem?
4. How does a flowchart proof structure help in organizing a proof?
5. Can the Congruent Supplements Theorem apply to obtuse angles?

**Tip**: Always check if the given angles form a linear pair (i.e., are adjacent and sum to \( 180^\circ \)) when dealing with supplementary angles.

Prove the Congruent Supplements Theorem using the given diagram and statements: ∠HOD ≅ ∠ZUR. Prove that ∠DOC ≅ ∠RUJ.

Learn how to prove the Congruent Supplements Theorem with a detailed step-by-step solution. This problem covers key geometry concepts including congruent angles, supplementary angles, and the Congruent Supplements Theorem. Suitable for students in Grades 8-10.

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