https://file.aiquickdraw.com/mathbot/file/1729547875932s3abdzjw.jpg

Let's analyze and work through the three proofs that are shown in the image. Here's how we can approach each one.

### Problem 9
**Given:**
- \( U \) is the midpoint of \( ST \)
- \( SV \cong TW \)
- \( VU = WU \)

**To Prove:**  
\( \triangle SVU \cong \triangle TWU \)

#### Statements and Reasons:
1. **Statement:** \( U \) is the midpoint of \( ST \)  
   **Reason:** Given

2. **Statement:** \( SV \cong TW \)  
   **Reason:** Given

3. **Statement:** \( VU = WU \)  
   **Reason:** Given

4. **Statement:** \( SU = UT \)  
   **Reason:** Definition of midpoint (midpoint splits a segment into two equal parts)

5. **Statement:** \( \triangle SVU \cong \triangle TWU \)  
   **Reason:** CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

### Problem 10
This section is not yet filled in. The given information involves parallel lines and congruent angles, and we can deduce how to proceed.

**Given:**
- \( MN \parallel PO \)
- \( MP \parallel NO \)

**To Prove:**  
\( MP \cong ON \)

You can solve this by showing that \( \triangle MPN \cong \triangle ONP \) using criteria such as the properties of parallel lines, alternate interior angles, and possibly the reflexive property of \( NP \).

### Problem 11
**Given:**
- \( DE \parallel FG \)
- \( DE \cong FG \)
- \( \angle DEF \cong \angle FGH \)

**To Prove:**  
\( \angle DFE \cong \angle FHG \)

#### Statements and Reasons:
1. **Statement:** \( DE \parallel FG \)  
   **Reason:** Given

2. **Statement:** \( DE \cong FG \)  
   **Reason:** Given

3. **Statement:** \( \angle DEF \cong \angle FGH \)  
   **Reason:** Given

4. **Statement:** Corresponding angles are congruent  
   **Reason:** Corresponding angles are equal when two lines are parallel.

5. **Statement:** \( \angle DFE \cong \angle FHG \)  
   **Reason:** Since corresponding angles are congruent, this follows from the parallel lines and angle congruence.

Would you like further details on any of these problems?

Here are five related questions to expand your understanding:

1. What is the difference between congruent triangles and similar triangles?
2. How does the midpoint theorem help in proving triangle congruence?
3. How can we use alternate interior angles in parallel line proofs?
4. What role does the CPCTC principle play in geometric proofs?
5. How can we use the properties of parallelograms in proving sides are congruent?

**Tip:** Always check whether the triangles share a common side or angle, as this can simplify the proof significantly!

The image contains three geometry proofs involving congruent triangles and parallel lines. Problem 9 asks to prove triangle congruence using midpoint and CPCTC. Problem 10 involves proving congruence between two triangles formed by parallel lines. Problem 11 is a proof of angle congruence using parallel lines and corresponding angles.

Solve geometry problems involving triangle congruence using the midpoint theorem, CPCTC, and parallel line properties. Explore how corresponding angles are used to prove congruence in problems with parallel lines. Suitable for Grades 9-10.

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