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The image provided contains a geometry proof where we are given that two sets of angles are supplementary, and we are tasked to prove that two angles are congruent.

### Given:
- \( \angle 1 \) and \( \angle 2 \) are supplementary.
- \( \angle 3 \) and \( \angle 2 \) are supplementary.

### Prove:
- \( \angle 1 \cong \angle 3 \).

### Statements and Reasons (Complete the Proof):

| Statements                              | Reasons                              |
|-----------------------------------------|--------------------------------------|
| \( m\angle 1 + m\angle 2 = 180 \)       | Given (definition of supplementary angles) |
| \( m\angle 3 + m\angle 2 = 180 \)       | Given (definition of supplementary angles) |
| \( m\angle 1 + m\angle 2 = m\angle 3 + m\angle 2 \) | Substitution (from both supplementary conditions) |
| \( m\angle 1 = m\angle 3 \)             | Subtraction Property of Equality (cancel out \( m\angle 2 \)) |
| \( \angle 1 \cong \angle 3 \)           | Definition of Congruence (equal measures imply congruence) |

The proof steps involve recognizing that both pairs of angles involving \( \angle 2 \) sum to 180°, substituting, and then using the subtraction property of equality to cancel out \( m\angle 2 \), showing that the measures of \( \angle 1 \) and \( \angle 3 \) are equal, which means the angles are congruent.

### Do you have any questions or would you like further details?

### Related Questions:
1. What does it mean for angles to be supplementary?
2. Can two angles be congruent and supplementary at the same time?
3. How does the substitution property help in geometric proofs?
4. What other properties of equality are commonly used in geometry?
5. How is the definition of congruence different from the definition of equality in angles?

**Tip:** When doing geometry proofs, always keep track of what you're given and what needs to be proven, as this will guide the logical steps you need to take.

Given: ∠1 and ∠2 are supplementary and ∠3 and ∠2 are supplementary. Prove: ∠1 ≅ ∠3.

Learn how to prove angle congruence using supplementary angles in this geometry proof. This problem demonstrates key concepts such as the definition of supplementary angles, substitution, and the subtraction property of equality. Suitable for students in Grades 8-10.

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