This geometry worksheet provides structured practice with the segment addition postulate through eight problems that progress from simple calculations to algebraic equations.

Problem Analysis and Solution

The worksheet involves using the Segment Addition Postulate, which states that if a point \( M \) lies on a segment \( \overline{LN} \), then:
\[
LM + MN = LN
\]
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:
\[
LN = 22 + 15 = 37
\]

Answer:
\[
\boxed{37}
\]

---

#### Problem 2:
If \( LN = 54 \) and \( LM = 31 \), find \( MN \).

Solution:
Using the Segment Addition Postulate:
\[
LN = LM + MN
\]
Rearrange to solve for \( MN \):
\[
MN = LN - LM
\]
Substitute the given values:
\[
MN = 54 - 31 = 23
\]

Answer:
\[
\boxed{23}
\]

---

#### Problem 3:
If \( RT = 36 \), find the value of \( x \).

Diagram:
\[
R \quad S \quad T
\]
\[
RS = 6x + 1, \quad ST = x + 7
\]

Solution:
Using the Segment Addition Postulate:
\[
RT = RS + ST
\]
Substitute the given expressions:
\[
36 = (6x + 1) + (x + 7)
\]
Simplify:
\[
36 = 6x + 1 + x + 7
\]
\[
36 = 7x + 8
\]
Isolate \( x \):
\[
36 - 8 = 7x
\]
\[
28 = 7x
\]
\[
x = 4
\]

Answer:
\[
\boxed{4}
\]

---

#### Problem 4:
If \( DF = 9x - 39 \), find \( EF \).

Diagram:
\[
D \quad E \quad F
\]
\[
DE = 47, \quad EF = 3x + 10
\]

Solution:
Using the Segment Addition Postulate:
\[
DF = DE + EF
\]
Substitute the given expressions:
\[
9x - 39 = 47 + (3x + 10)
\]
Simplify:
\[
9x - 39 = 47 + 3x + 10
\]
\[
9x - 39 = 3x + 57
\]
Isolate \( x \):
\[
9x - 3x = 57 + 39
\]
\[
6x = 96
\]
\[
x = 16
\]
Now, find \( EF \):
\[
EF = 3x + 10 = 3(16) + 10 = 48 + 10 = 58
\]

Answer:
\[
\boxed{58}
\]

---

#### Problem 5:
If \( UW = 6x - 35 \), find \( UW \).

Diagram:
\[
U \quad V \quad W
\]
\[
UV = 19, \quad VW = 4x - 20
\]

Solution:
Using the Segment Addition Postulate:
\[
UW = UV + VW
\]
Substitute the given expressions:
\[
6x - 35 = 19 + (4x - 20)
\]
Simplify:
\[
6x - 35 = 19 + 4x - 20
\]
\[
6x - 35 = 4x - 1
\]
Isolate \( x \):
\[
6x - 4x = -1 + 35
\]
\[
2x = 34
\]
\[
x = 17
\]
Now, find \( UW \):
\[
UW = 6x - 35 = 6(17) - 35 = 102 - 35 = 67
\]

Answer:
\[
\boxed{67}
\]

---

#### Problem 6:
If \( HJ = 7x - 27 \), find the value of \( x \).

Diagram:
\[
H \quad I \quad J
\]
\[
HI = 3x - 5, \quad IJ = x - 1
\]

Solution:
Using the Segment Addition Postulate:
\[
HJ = HI + IJ
\]
Substitute the given expressions:
\[
7x - 27 = (3x - 5) + (x - 1)
\]
Simplify:
\[
7x - 27 = 3x - 5 + x - 1
\]
\[
7x - 27 = 4x - 6
\]
Isolate \( x \):
\[
7x - 4x = -6 + 27
\]
\[
3x = 21
\]
\[
x = 7
\]

Answer:
\[
\boxed{7}
\]

---

#### Problem 7:
If \( BD = 7x - 10 \), \( BC = 4x - 29 \), and \( CD = 5x - 9 \), find each value.

Diagram:
\[
B \quad C \quad D
\]

Solution:
Using the Segment Addition Postulate:
\[
BD = BC + CD
\]
Substitute the given expressions:
\[
7x - 10 = (4x - 29) + (5x - 9)
\]
Simplify:
\[
7x - 10 = 4x - 29 + 5x - 9
\]
\[
7x - 10 = 9x - 38
\]
Isolate \( x \):
\[
7x - 9x = -38 + 10
\]
\[
-2x = -28
\]
\[
x = 14
\]
Now, find \( BC \), \( CD \), and \( BD \):
\[
BC = 4x - 29 = 4(14) - 29 = 56 - 29 = 27
\]
\[
CD = 5x - 9 = 5(14) - 9 = 70 - 9 = 61
\]
\[
BD = 7x - 10 = 7(14) - 10 = 98 - 10 = 88
\]

Answers:
\[
x = \boxed{14}, \quad BC = \boxed{27}, \quad CD = \boxed{61}, \quad BD = \boxed{88}
\]

---

#### Problem 8:
If \( \overline{BD} \cong \overline{BC} \), \( BD = 5x - 26 \), \( BC = 2x + 1 \), and \( AC = 43 \), find \( AB \).

Diagram:
\[
A \quad B \quad C \quad D
\]

Solution:
Since \( \overline{BD} \cong \overline{BC} \), their lengths are equal:
\[
BD = BC
\]
Substitute the given expressions:
\[
5x - 26 = 2x + 1
\]
Isolate \( x \):
\[
5x - 2x = 1 + 26
\]
\[
3x = 27
\]
\[
x = 9
\]
Now, find \( BD \) and \( BC \):
\[
BD = 5x - 26 = 5(9) - 26 = 45 - 26 = 19
\]
\[
BC = 2x + 1 = 2(9) + 1 = 18 + 1 = 19
\]
Since \( BD = BC = 19 \), we use the Segment Addition Postulate for \( AC \):
\[
AC = AB + BC
\]
Substitute the known values:
\[
43 = AB + 19
\]
Solve for \( AB \):
\[
AB = 43 - 19 = 24
\]

Answer:
\[
\boxed{24}
\]

---

Final Answers:

1. \(\boxed{37}\)
2. \(\boxed{23}\)
3. \(\boxed{4}\)
4. \(\boxed{58}\)
5. \(\boxed{67}\)
6. \(\boxed{7}\)
7. \(x = \boxed{14}\), \(BC = \boxed{27}\), \(CD = \boxed{61}\), \(BD = \boxed{88}\)
8. \(\boxed{24}\)