https://file.aiquickdraw.com/mathbot/file/17297396814510pbwy184.jpg

The diagram and the problem suggest that we are working with a segment addition problem. We are given:

- \( LM = 9 \)
- \( LP = 20 \)

We need to prove that \( MP = 11 \).

### Solution steps:

We can use the **Segment Addition Postulate**, which states that if point \( M \) is between points \( L \) and \( P \), then:

\[
LM + MP = LP
\]

Substitute the given values:

\[
9 + MP = 20
\]

Solve for \( MP \):

\[
MP = 20 - 9 = 11
\]

Thus, we have proven that \( MP = 11 \).

### Proof:
| Statements        | Reasons                       |
|-------------------|-------------------------------|
| 1. \( LM = 9 \), \( LP = 20 \) | 1. Given |
| 2. \( LM + MP = LP \)          | 2. Segment Addition Postulate |
| 3. \( 9 + MP = 20 \)           | 3. Substitution |
| 4. \( MP = 11 \)               | 4. Solving the equation |

Would you like further clarification on this proof or additional examples?

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Here are 5 related questions to expand on the concept:
1. What is the Segment Addition Postulate?
2. How would the solution change if \( LM \) and \( LP \) were given as variables?
3. Can the Segment Addition Postulate be applied to geometric figures other than line segments?
4. How do we verify if a point truly lies between two other points on a segment?
5. What is the significance of segment congruence in geometry proofs?

**Tip:** When solving geometry problems, always check if you can apply basic postulates like the Segment Addition Postulate or the Angle Addition Postulate to simplify your reasoning.

This proof demonstrates how to apply the Segment Addition Postulate to show that MP = 11, given that LM = 9 and LP = 20. Suitable for students in Grades 6-8, this step-by-step solution helps in understanding basic geometric concepts.

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