https://file.aiquickdraw.com/mathbot/file/1726348236102p61qz5al.jpg

Let's fill out the proof in the two-column format based on the given theorem and information.

### Theorem: 
If two angles are supplementary to the same angle, then the two angles are congruent.

### Given: 
- ∠R and ∠S are each supplementary to ∠T.

### Prove: 
- ∠R ≅ ∠S

#### Two-column proof:

| **Statements**                                     | **Reasons**                                      |
|----------------------------------------------------|--------------------------------------------------|
| 1. ∠R and ∠S are supplementary to ∠T.             | 1. Given                                         |
| 2. ∠R + ∠T = 180° and ∠S + ∠T = 180°.             | 2. Definition of supplementary angles            |
| 3. ∠R + ∠T = ∠S + ∠T                              | 3. Substitution (both equal 180°)                |
| 4. Subtract ∠T from both sides: ∠R = ∠S           | 4. Subtraction Property of Equality              |
| 5. ∠R ≅ ∠S                                        | 5. Definition of congruent angles (equal measure)|

This concludes the proof that ∠R is congruent to ∠S.

Would you like further details or explanations on any steps? 

Here are 5 related questions to expand understanding:
1. What does it mean for two angles to be supplementary?
2. Can two obtuse angles be supplementary?
3. How does the definition of congruent angles relate to equality of their measures?
4. What is the significance of the Subtraction Property of Equality in this proof?
5. How would the proof change if ∠R and ∠S were complementary instead of supplementary?

**Tip**: When working on geometric proofs, always ensure that each statement has a logical reason to help maintain clarity and rigor.

Theorem: If two angles are supplementary to the same angle, then the two angles are congruent. Given: ∠R and ∠S are each supplementary to ∠T. Prove: ∠R ≅ ∠S.

Learn how to prove that two angles are congruent if they are supplementary to the same angle. This proof utilizes key geometric concepts like the definition of supplementary and congruent angles. Suitable for students in Grades 9-10.

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