https://file.aiquickdraw.com/mathbot/file/17274709702939rftnt1n.jpg

This image depicts a two-column proof setup, where the left column lists the statements and the right column contains the reasons. The goal is to prove that \( 2 \times \text{m} \angle LMP = \text{m} \angle LMN \), with the given information that \( MP \) bisects \( \angle LMN \).

Let's go through the steps to complete the proof:

### **Statements and Reasons:**

1. **Statement**: \( MP \) bisects \( \angle LMN \).
   - **Reason**: Given.

2. **Statement**: \( \angle LMP \cong \angle NMP \).
   - **Reason**: Definition of an angle bisector. If a ray (like \( MP \)) bisects an angle, it divides the angle into two congruent parts.

3. **Statement**: \( \text{m} \angle LMP + \text{m} \angle NMP = \text{m} \angle LMN \).
   - **Reason**: Angle Addition Postulate. The sum of the two smaller angles created by the bisector equals the entire angle.

4. **Statement**: \( \text{m} \angle LMP = \text{m} \angle NMP \).
   - **Reason**: Definition of congruent angles. If two angles are congruent, their measures are equal.

5. **Statement**: \( \text{m} \angle LMP + \text{m} \angle LMP = \text{m} \angle LMN \).
   - **Reason**: Substitution Property of Equality. Since \( \text{m} \angle LMP = \text{m} \angle NMP \), we can substitute \( \text{m} \angle LMP \) for \( \text{m} \angle NMP \).

6. **Statement**: \( 2 \times \text{m} \angle LMP = \text{m} \angle LMN \).
   - **Reason**: Distributive Property. Adding two equal angles is the same as multiplying one of them by 2.

Thus, the proof is completed by following these logical steps.

Would you like further details on any step, or have questions about the process?

### 5 Related Questions:
1. What is the Angle Addition Postulate and how is it applied in this proof?
2. How does the definition of an angle bisector help in solving angle-based proofs?
3. Why is the Substitution Property of Equality important in geometric proofs?
4. Can you provide an example where the Distributive Property is used in a different type of geometric proof?
5. How can congruent angles be identified in more complex geometric figures?

### Tip:
When working on geometry proofs, always start with the given information and build your reasoning logically step by step to avoid mistakes.

Prove that 2(m∠LMP) = m∠LMN given that MP bisects ∠LMN.

In this geometry proof, learn how to prove that 2(m∠LMP) = m∠LMN using key concepts like angle bisectors, the Angle Addition Postulate, and congruent angles. Step-by-step reasoning suitable for students in Grades 9-11.

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