Problem Analysis:
The image contains problems related to the
Segment Addition Postulate, which states that if point \( M \) is between points \( L \) and \( N \), then:
\[
LN = LM + MN
\]
We will solve each problem step by step.
---
Problem 1:
If \( LM = 22 \) and \( MN = 15 \), find \( LN \).
#### Solution:
Using the Segment Addition Postulate:
\[
LN = LM + MN
\]
Substitute the given values:
\[
LN = 22 + 15
\]
\[
LN = 37
\]
#### Final Answer:
\[
\boxed{37}
\]
---
Problem 2:
If \( LN = 54 \) and \( LM = 31 \), find \( MN \).
#### Solution:
Using the Segment Addition Postulate:
\[
LN = LM + MN
\]
Rearrange the equation to solve for \( MN \):
\[
MN = LN - LM
\]
Substitute the given values:
\[
MN = 54 - 31
\]
\[
MN = 23
\]
#### Final Answer:
\[
\boxed{23}
\]
---
Problem 4:
If \( DF = 9x - 39 \), find \( EF \).
#### Solution:
From the diagram, we see that point \( E \) is between points \( D \) and \( F \). Therefore, using the Segment Addition Postulate:
\[
DF = DE + EF
\]
From the diagram, we are given:
\[
DE = 47 \quad \text{and} \quad EF = 3x + 10
\]
Substitute these into the equation:
\[
DF = DE + EF
\]
\[
9x - 39 = 47 + (3x + 10)
\]
Simplify the right-hand side:
\[
9x - 39 = 47 + 3x + 10
\]
\[
9x - 39 = 57 + 3x
\]
Isolate \( x \) by subtracting \( 3x \) from both sides:
\[
9x - 3x - 39 = 57
\]
\[
6x - 39 = 57
\]
Add 39 to both sides:
\[
6x = 96
\]
Divide by 6:
\[
x = 16
\]
Now, substitute \( x = 16 \) back into the expression for \( EF \):
\[
EF = 3x + 10
\]
\[
EF = 3(16) + 10
\]
\[
EF = 48 + 10
\]
\[
EF = 58
\]
#### Final Answer:
\[
\boxed{58}
\]
---
Summary of Answers:
1. \( LN = \boxed{37} \)
2. \( MN = \boxed{23} \)
4. \( EF = \boxed{58} \)
Parent Tip: Review the logic above to help your child master the concept of geometry basics worksheet answers.
