Geometry worksheet focusing on angle theorems and proofs.

It looks like you've uploaded an image of a geometry worksheet focused on angle relationships, particularly involving supplementary angles, linear pairs, and the Congruent Supplements Theorem. The worksheet includes problems that require students to prove angle relationships using given information and geometric theorems.

Let’s walk through the problems step by step, as they appear in your image (based on the visible content):

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Problem 1: Prove that ∠1 ≅ ∠3

#### Given:
- ∠1 and ∠2 are supplementary
- ∠3 and ∠2 are supplementary

#### Prove:
- ∠1 ≅ ∠3

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#### Proof (Two-Column Format)

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

✔ Conclusion: ∠1 ≅ ∠3 by the Congruent Supplements Theorem — if two angles are supplementary to the same angle, then they are congruent.

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Problem 2: Prove that ∠1 ≅ ∠3

#### Given:
- ∠1 and ∠2 are supplementary
- ∠3 and ∠2 are supplementary

This is identical to Problem 1 — just a different diagram (likely showing intersecting lines or a linear pair setup). The proof follows exactly the same logic.

So, again:

| Statements | Reasons |
|----------------|-------------|
| 1. ∠1 and ∠2 are supplementary | 1. Given |
| 2. ∠3 and ∠2 are supplementary | 2. Given |
| 3. m∠1 + m∠2 = 180° | 3. Definition of Supplementary Angles |
| 4. m∠3 + m∠2 = 180° | 4. Definition of Supplementary Angles |
| 5. m∠1 + m∠2 = m∠3 + m∠2 | 5. Transitive Property |
| 6. m∠1 = m∠3 | 6. Subtraction Property |
| 7. ∠1 ≅ ∠3 | 7. Definition of Congruent Angles |

✔ Same result: ∠1 ≅ ∠3

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Problem 3: Prove that ∠1 ≅ ∠3

#### Given:
- ∠1 and ∠2 are supplementary
- ∠3 and ∠2 are supplementary

Again, this is the same structure as before. So the proof is repeated.

But wait — perhaps the diagram shows something more complex? Let's consider what might be going on.

Looking at the diagrams:
- In Problem 1, it shows two angles sharing a common side (like a linear pair).
- In Problem 2, it shows two angles adjacent to the same angle.
- In Problem 3, it may show two intersecting lines, forming vertical angles and linear pairs.

Wait — actually, let's look carefully at the diagram in Problem 3:

> It shows two lines intersecting, forming four angles: ∠1, ∠2, ∠3, ∠4.

From the labeling:
- ∠1 and ∠2 form a linear pair
- ∠3 and ∠2 also form a linear pair?
- But actually, likely: ∠1 and ∠2 are adjacent, ∠3 and ∠2 are adjacent but on the other side?

Wait — no, in standard notation:
- If two lines intersect, opposite angles are vertical angles.
- Adjacent angles are linear pairs (sum to 180°).

Let’s suppose:
- ∠1 and ∠2 are adjacent (linear pair), so supplementary
- ∠3 and ∠2 are also adjacent? That would mean ∠3 is on the other side of ∠2.

But that can't be unless ∠1 and ∠3 are vertical angles.

Ah! Here’s the key:

In Problem 3, we’re probably being asked to prove that vertical angles are congruent, using the Congruent Supplements Theorem.

Let’s re-analyze with correct assumptions.

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Revised Problem 3: Prove ∠1 ≅ ∠3 (Vertical Angles)

#### Given:
- Lines AB and CD intersect at point E
- ∠1 and ∠2 are supplementary (linear pair)
- ∠3 and ∠2 are supplementary (linear pair)

#### Prove:
- ∠1 ≅ ∠3

This is a classic proof of vertical angles are congruent.

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#### Proof

| Statements | Reasons |
|----------------|-------------|
| 1. ∠1 and ∠2 are supplementary | 1. Given (linear pair) |
| 2. ∠3 and ∠2 are supplementary | 2. Given (linear pair) |
| 3. m∠1 + m∠2 = 180° | 3. Definition of Supplementary Angles |
| 4. m∠3 + m∠2 = 180° | 4. Definition of Supplementary Angles |
| 5. m∠1 + m∠2 = m∠3 + m∠2 | 5. Transitive Property |
| 6. m∠1 = m∠3 | 6. Subtraction Property of Equality |
| 7. ∠1 ≅ ∠3 | 7. Definition of Congruent Angles |

✔ This proves that vertical angles are congruent via the Congruent Supplements Theorem.

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✔ Summary of Key Concepts Used:

1. Supplementary Angles: Two angles whose measures add up to 180°.
2. Linear Pair Theorem: If two angles form a linear pair, they are supplementary.
3. Congruent Supplements Theorem: If two angles are supplementary to the same angle, then they are congruent.
4. Transitive Property of Equality: If a = b and b = c, then a = c.
5. Subtraction Property of Equality: If a + b = c + b, then a = c.
6. Definition of Congruent Angles: Angles with equal measures.

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Final Answer:

All three problems use the same logical structure:

> If two angles are supplementary to the same angle, then they are congruent.

Thus, ∠1 ≅ ∠3 in each case, based on the Congruent Supplements Theorem.

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📌 Important Note:

The Congruent Supplements Theorem is often used to prove that vertical angles are congruent. Even though the problem doesn’t explicitly say “vertical angles,” the diagram and given conditions imply that ∠1 and ∠3 are vertical angles, each supplementary to ∠2.

So the core idea is:
> Vertical angles are congruent because they are supplements of the same angle.

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