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This image shows a two-column geometric proof, with "Statements" on the left side and "Reasons" on the right side. The task is to fill in the blanks in both columns by logically matching statements to their correct reasons using the provided options at the bottom.

### Let's work through the proof:

#### Step-by-step:
1. **Statement 1:**
   - Given: ∠1 ≅ ∠3 and m∠3 = 90°
   - **Reason 1:** *Given*

2. **Statement 2:**
   - ∠1 ≅ ∠2, ∠3 ≅ ∠4 (using congruent angles)
   - **Reason 2:** *Definition of congruent angles*

3. **Statement 3:**
   - m∠1 = 90°
   - **Reason 3:** *Substitution* (m∠1 ≅ m∠3, and m∠3 = 90°)

4. **Statement 4:**
   - ∠1 ≅ ∠2 (congruent vertical angles)
   - **Reason 4:** *Vertical Angles Theorem*

5. **Statement 5:**
   - m∠1 = m∠2 and m∠3 = m∠4
   - **Reason 5:** *Substitution*

6. **Statement 6:**
   - m∠2 = 90°, m∠4 = 90°
   - **Reason 6:** *Substitution*

7. **Statement 7:**
   - m∠2 + m∠4 = 180°
   - **Reason 7:** *Angle Addition Postulate*

8. **Statement 8:**
   - ∠2 and ∠4 are supplementary
   - **Reason 8:** *Definition of supplementary angles*

### Proof Solution:

1. **Statements:**
   1. ∠1 ≅ ∠3 and m∠3 = 90°
   2. ∠1 ≅ ∠2, ∠3 ≅ ∠4
   3. m∠1 = 90°
   4. ∠1 ≅ ∠2
   5. m∠1 = m∠2 and m∠3 = m∠4
   6. m∠2 = 90°, m∠4 = 90°
   7. m∠2 + m∠4 = 180°
   8. ∠2 and ∠4 are supplementary

2. **Reasons:**
   1. Given
   2. Definition of congruent angles
   3. Substitution
   4. Vertical Angles Theorem
   5. Substitution
   6. Substitution
   7. Angle Addition Postulate
   8. Definition of supplementary angles

Would you like further details, or have any questions?

Here are 5 related questions to explore this topic:
1. What is the Vertical Angles Theorem?
2. How do you define supplementary angles?
3. What is the difference between substitution and congruence in proofs?
4. How does the Angle Addition Postulate work in other geometric problems?
5. How do you prove angles are congruent using other postulates?

**Tip:** Always start with given information and definitions when solving geometric proofs, as they often lead you to the correct logical steps.

Proof: Fill in the missing statements and reasons involving congruent angles, angle measures, and supplementary angles.

Explore a detailed geometry proof involving congruent angles, vertical angles, and supplementary angles. Learn how to apply the Vertical Angles Theorem, substitution, and the Angle Addition Postulate. Suitable for students in Grades 9-10.

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