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To complete the proof that ∠HJI ≅ ∠FJG, here's the step-by-step reasoning for each statement:

1. **Statement 1**: \( m\angle FJG + m\angle GJH = 180^\circ \)
   - **Reason**: These are **angles forming a linear pair**, which means they sum to \( 180^\circ \) because they are on a straight line.

2. **Statement 2**: \( m\angle GJH + m\angle HJI = 180^\circ \)
   - **Reason**: Similarly, **angles forming a linear pair** sum to \( 180^\circ \). This applies to the angles on the opposite side of the intersection.

3. **Statement 3**: \( m\angle FJG + m\angle GJH = m\angle GJH + m\angle HJI \)
   - **Reason**: This follows from the **Transitive Property of Equality**. Since both sides equal \( 180^\circ \), you can set the two equations equal to each other.

4. **Statement 4**: \( m\angle FJG = m\angle HJI \)
   - **Reason**: By **subtracting** \( m\angle GJH \) from both sides of the previous equation, we isolate the two angles \( \angle FJG \) and \( \angle HJI \).

5. **Statement 5**: \( \angle HJI \cong \angle FJG \)
   - **Reason**: **Definition of congruent angles**. Since their measures are equal, the angles are congruent.

This completes the proof that \( \angle HJI \cong \angle FJG \).

Let me know if you'd like more details or clarifications! 

Here are some follow-up questions to explore:
1. What does it mean for two angles to form a linear pair?
2. Why is the Transitive Property useful in this proof?
3. How do we define congruent angles?
4. What is the relationship between supplementary angles and linear pairs?
5. Can you think of another example of congruent angles formed in an intersection like this?

**Tip**: Always make sure to identify whether angles form a linear pair or are vertical angles when solving geometry proofs.

Complete the proof that ∠HJI ≅ ∠FJG.

Explore a detailed step-by-step geometry proof to demonstrate that angles ∠HJI and ∠FJG are congruent. This problem utilizes the Linear Pair Postulate and the Transitive Property of Equality, essential for understanding angle relationships in Geometry. Suitable for students in Grades 8-10.

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