Worksheet for practicing two-column proofs in geometry, covering key theorems and angle relationships.

It looks like you've uploaded an image of a geometry worksheet that involves identifying and applying geometric theorems and postulates related to linear pairs, supplementary angles, vertical angles, and congruent angles. The task is to complete each section by filling in the correct reasons or statements based on the given information.

Let's go through each part step-by-step and solve it with explanations.

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Part 1: Complete the following using the Linear Pair Theorem

> "If ∠1 and ∠2 form a linear pair, then they are supplementary."

#### Given:
- ∠1 and ∠2 form a linear pair.

#### To Prove:
- ∠1 and ∠2 are supplementary.

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Step-by-step Solution:

| Statement | Reason |
|---------|--------|
| 1. ∠1 and ∠2 form a linear pair | Given |
| 2. ∠1 and ∠2 are adjacent angles | Definition of Linear Pair (adjacent angles that form a straight line) |
| 3. m∠1 + m∠2 = 180° | Angle Addition Postulate (they form a straight angle) |
| 4. ∠1 and ∠2 are supplementary | Definition of Supplementary Angles (sum to 180°) |

✔ So, the Linear Pair Theorem states: *If two angles form a linear pair, then they are supplementary.*

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Part 2: Write a proof of the Supplemental Theorem

> "If ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary, then ∠1 ≅ ∠3."

This is the Congruent Supplements Theorem.

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Given:
- ∠1 and ∠2 are supplementary → m∠1 + m∠2 = 180°
- ∠2 and ∠3 are supplementary → m∠2 + m∠3 = 180°

To Prove:
- ∠1 ≅ ∠3 (i.e., m∠1 = m∠3)

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Proof:

| Statement | Reason |
|---------|--------|
| 1. ∠1 and ∠2 are supplementary | Given |
| 2. ∠2 and ∠3 are supplementary | Given |
| 3. m∠1 + m∠2 = 180° | Definition of Supplementary Angles |
| 4. m∠2 + m∠3 = 180° | Definition of Supplementary Angles |
| 5. m∠1 + m∠2 = m∠2 + m∠3 | Substitution Property of Equality (both equal 180°) |
| 6. m∠1 = m∠3 | Subtraction Property of Equality (subtract m∠2 from both sides) |
| 7. ∠1 ≅ ∠3 | Definition of Congruent Angles (angles with equal measures) |

✔ This proves the Congruent Supplements Theorem: *If two angles are supplementary to the same angle, then they are congruent.*

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Part 3: Write a proof of the Vertical Angles Congruence Theorem

> "If ∠1 and ∠2 are vertical angles, then ∠1 ≅ ∠2."

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Given:
- ∠1 and ∠2 are vertical angles.

To Prove:
- ∠1 ≅ ∠2

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Proof:

| Statement | Reason |
|---------|--------|
| 1. ∠1 and ∠2 are vertical angles | Given |
| 2. ∠1 and ∠3 form a linear pair | Definition of Linear Pair (assume ∠3 is adjacent to ∠1) |
| 3. ∠2 and ∠3 form a linear pair | Definition of Linear Pair |
| 4. ∠1 and ∠3 are supplementary | Linear Pair Theorem |
| 5. ∠2 and ∠3 are supplementary | Linear Pair Theorem |
| 6. m∠1 + m∠3 = 180° | Definition of Supplementary Angles |
| 7. m∠2 + m∠3 = 180° | Definition of Supplementary Angles |
| 8. m∠1 + m∠3 = m∠2 + m∠3 | Substitution (both equal 180°) |
| 9. m∠1 = m∠2 | Subtraction Property of Equality |
| 10. ∠1 ≅ ∠2 | Definition of Congruent Angles |

✔ This proves the Vertical Angles Congruence Theorem: *Vertical angles are congruent.*

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Summary of Key Theorems Used:

| Theorem | Description |
|-------|-----------|
| Linear Pair Theorem | If two angles form a linear pair, then they are supplementary. |
| Congruent Supplements Theorem | If two angles are supplementary to the same angle, then they are congruent. |
| Vertical Angles Congruence Theorem | Vertical angles are congruent. |

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Final Notes:

- The red markings in your image likely indicate corrections or answers already filled in.
- Your goal was to fill in the missing statements and reasons in each proof.
- All proofs rely on definitions, postulates (like Angle Addition), and properties of equality.

If you'd like me to format this as a clean printable version or help with any specific blank you're stuck on, let me know!