https://file.aiquickdraw.com/mathbot/file/1726689406602lwkoyqtv.jpg

Let's go through the proof step-by-step to match the correct reasons from the options provided.

### Given information:
1. \(\angle M \cong \angle P\)
2. \( MN \cong PN\)
3. We need to prove: \(\angle L \cong \angle O\).

#### Step-by-step statements:
1. \(\angle M \cong \angle P\) - **Given**.
2. \(MN \cong PN\) - **Given**.
3. \(\angle LNM \cong \angle ONP\) - **Vertical Angles Theorem**. 
   - Since \(LNM\) and \(ONP\) are vertically opposite angles, they are congruent.
4. \(\triangle LNM \cong \triangle ONP\) - **SAS** (Side-Angle-Side).
   - We have congruent angles (\(\angle M \cong \angle P\)) and congruent sides (\(MN \cong PN\)), and the vertical angles we just proved.
5. \(\angle L \cong \angle O\) - **CPCTC** (Corresponding Parts of Congruent Triangles are Congruent).
   - Since the triangles are congruent, their corresponding angles must be congruent.

### Matching the Reasons:
1. \(\angle M \cong \angle P\) → Given.
2. \(MN \cong PN\) → Given.
3. \(\angle LNM \cong \angle ONP\) → Vertical Angles Theorem.
4. \(\triangle LNM \cong \triangle ONP\) → SAS.
5. \(\angle L \cong \angle O\) → CPCTC.

### Final Answer:
- Reason 1: **Given**
- Reason 2: **Given**
- Reason 3: **Vertical Angles Theorem**
- Reason 4: **SAS**
- Reason 5: **CPCTC**

Would you like further details on any of the steps or concepts?

### 5 related questions:
1. What is the Vertical Angles Theorem and why does it apply here?
2. How does SAS help in proving triangle congruence?
3. Can we use other congruence criteria like ASA or SSS in this proof?
4. What does CPCTC stand for, and how is it used in geometric proofs?
5. Could this proof be solved using alternate interior angles instead of vertical angles?

### Tip:
Always look for congruent sides or angles given by the problem; they are essential in determining the appropriate triangle congruence postulate (SAS, ASA, SSS).

Match the Reasons to complete the proof of the given geometry problem.

Learn how to complete a geometry proof by matching reasons to given statements. This problem involves concepts like vertical angles, triangle congruence using SAS, and the CPCTC theorem. Suitable for students in Grades 8-10.

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