Vertical Angles and Angle Sum Theorem Proofs worksheet with answers.

Actually, I can’t see or process images — including the one you uploaded — because I don’t have image recognition or file-reading capabilities. However, based on the text you provided, which appears to be a worksheet titled:

> Vertical Angles & Angle Sum Theorem Proofs (Leveled)

I can help you solve and explain all three proofs shown in the text. Let’s go step by step.

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## 🔹 PROOF 1: Vertical Angles are Congruent
*(Given: Lines intersect forming vertical angles ∠1 and ∠3)*

Goal: Prove that ∠1 ≅ ∠3

Statements & Reasons:

| Statement | Reason |
|----------|--------|
| 1. ∠1 and ∠2 form a linear pair | Given (they’re adjacent and form a straight line) |
| 2. ∠2 and ∠3 form a linear pair | Given |
| 3. m∠1 + m∠2 = 180° | Definition of Linear Pair (supplementary angles) |
| 4. m∠2 + m∠3 = 180° | Definition of Linear Pair |
| 5. m∠1 + m∠2 = m∠2 + m∠3 | Transitive Property (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) |

✔ Conclusion: Vertical angles are congruent.

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## 🔹 PROOF 2: Triangle Angle Sum Theorem
*(Given: Triangle ABC with angles labeled ∠A, ∠B, ∠C)*

Goal: Prove that m∠A + m∠B + m∠C = 180°

*(Note: This proof typically uses a line drawn through one vertex parallel to the opposite side. Since the diagram isn't visible, I’ll use the standard approach.)*

Statements & Reasons:

| Statement | Reason |
|----------|--------|
| 1. Draw line ℓ through point A parallel to segment BC | Construction (we can always draw such a line) |
| 2. ∠1 ≅ ∠B, ∠2 ≅ ∠C | Alternate Interior Angles Theorem (since ℓ ∥ BC and AB, AC are transversals) |
| 3. ∠1 + ∠A + ∠2 = 180° | Definition of Straight Angle (angles on line ℓ at point A form a straight line) |
| 4. Substitute ∠B for ∠1 and ∠C for ∠2 | From Step 2 (congruent angles have equal measures) |
| 5. ∠B + ∠A + ∠C = 180° | Substitution Property |
| 6. m∠A + m∠B + m∠C = 180° | Rewriting using angle measures |

✔ Conclusion: The sum of the interior angles of any triangle is 180°.

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## 🔹 PROOF 3: Vertical Angles Again (Alternate Setup)
*(Given: Two lines intersect, forming vertical angles ∠1 and ∠3)*

This is essentially the same as Proof 1, but let’s complete it using the structure given:

Statements & Reasons:

| Statement | Reason |
|----------|--------|
| 1. ∠1 and ∠2 are supplementary | Definition of Linear Pair |
| 2. ∠2 and ∠3 are supplementary | Definition of Linear Pair |
| 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 | Transitive Property of Equality |
| 6. m∠1 = m∠3 | Subtraction Property of Equality |
| 7. ∠1 ≅ ∠3 | Definition of Congruent Angles |

✔ Same conclusion: Vertical angles are congruent.

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## 🧠 Summary of Key Concepts Used:

- Linear Pair → Supplementary Angles (sum to 180°)
- Transitive Property: If A = B and B = C, then A = C
- Subtraction Property: If A + C = B + C, then A = B
- Congruent Angles: Angles with equal measures
- Triangle Angle Sum Theorem: Uses parallel lines and alternate interior angles to show sum = 180°

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## ✔ Final Answer (Filled Worksheet Style):

Proof 1:

- 1. Given
- 2. Given
- 3. Definition of Linear Pair
- 4. Definition of Linear Pair
- 5. Transitive Property of Equality
- 6. Subtraction Property of Equality
- 7. Definition of Congruent Angles

Proof 2:

- 1. Through point A, draw line ℓ ∥ BC
- 2. ∠1 ≅ ∠B, ∠2 ≅ ∠C — Alternate Interior Angles Theorem
- 3. ∠1 + ∠A + ∠2 = 180° — Definition of Straight Angle
- 4. ∠B + ∠A + ∠C = 180° — Substitution
- 5. m∠A + m∠B + m∠C = 180° — Rewriting

*(Note: Some versions may label steps differently — adjust if your worksheet has specific labels.)*

Proof 3:

- 1. Definition of Linear Pair
- 2. Definition of Linear Pair
- 3. Definition of Supplementary Angles
- 4. Definition of Supplementary Angles
- 5. Transitive Property of Equality
- 6. Subtraction Property of Equality
- 7. Definition of Congruent Angles

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If you’d like me to format this as a printable answer key or adjust for a specific curriculum (like Common Core or Geometry Honors), just let me know!

Let me know if you want to explore why these theorems matter in real life or how they connect to other geometry concepts!