https://file.aiquickdraw.com/mathbot/file/17290553141101dg8fv25.jpg

Let's work through the proof in the image for problem 5.

### Problem Details:
- **Given:**
  - \( \angle FEH = \angle GFE = 90^\circ \)
  - \( EH \cong FG \)
- **To Prove:**
  - \( HF \cong EG \)

### Proof Structure:
We need to complete the two-column proof. Let's fill it in step by step:

#### Statements and Reasons:

1. **Statement:**
   \[
   \angle FEH = \angle GFE = 90^\circ, \ EH \cong FG
   \]
   **Reason:** Given.

2. **Statement:**
   \[
   \triangle FEH \cong \triangle GFE
   \]
   **Reason:** \( \text{Angle-Angle-Side (AAS) Theorem} \).  
   Since both triangles have:
   - \( \angle FEH = \angle GFE = 90^\circ \) (right angles),
   - \( EH \cong FG \) (given),
   - and the shared side \( EF \cong EF \) (reflexive property).

3. **Statement:**
   \[
   EF \cong FE
   \]
   **Reason:** Reflexive property (a side is congruent to itself).

4. **Statement:**
   \[
   \triangle HEF \cong \triangle GEF
   \]
   **Reason:** Side-Angle-Side (SAS) Theorem.

5. **Statement:**
   \[
   HF \cong GE
   \]
   **Reason:** Corresponding parts of congruent triangles (CPCTC).

---

This completes the proof! Do you need further clarification or details about any of the steps?

### Related Questions:
1. What are the key differences between AAS and SAS congruence theorems?
2. How does CPCTC work in proving congruence of triangle parts?
3. Can the reflexive property be applied to any shared sides in triangle congruence proofs?
4. Why is it important to identify the shared side when proving congruence?
5. In what situations would you use an angle-angle-side theorem over side-angle-side?

### Tip:
Always check for shared sides or angles in overlapping triangles when trying to prove congruence—this often simplifies your proof process!

Developing Proof. Complete the two-column proof: Given: angle FEH = angle GFE = 90 degrees, EH ≅ FG. Prove: HF ≅ EG.

Solve a triangle congruence proof using AAS and SAS theorems. The problem requires proving that HF ≅ EG given that angle FEH = angle GFE = 90 degrees and EH ≅ FG. Suitable for students in Grades 9-10.

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