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### Measuring Segments:
1. The distance between two points \( A \) and \( B \) is written as \( AB \), or \( d(A, B) \).

### Congruent Segments:
2. If two segments have the same length, then the segments are **congruent**. This is written as \( \overline{AB} \cong \overline{CD} \).

### Segment Addition Postulate:
3. If points \( A \), \( B \), and \( C \) are collinear and \( B \) is between \( A \) and \( C \), then \( AB + BC = AC \).

Now, let's solve the specific problems:

#### 1. If \( PQ = 9 \) and \( QR = 28 \), find \( PR \).
- \( PR = PQ + QR = 9 + 28 = 37 \).

#### 2. If \( QR = 17 \) and \( PR = 21 \), find \( PQ \).
- Using the segment addition postulate: \( PR = PQ + QR \). Therefore, \( PQ = PR - QR = 21 - 17 = 4 \).

#### 3. If \( EG = 71 \), find the value of \( x \). Given: \( 8x - 17 \) and \( 5x - 3 \).
- Since \( EG = EF + FG \):
  \[
  (8x - 17) + (5x - 3) = 71
  \]
  Simplifying:
  \[
  13x - 20 = 71 \quad \Rightarrow \quad 13x = 91 \quad \Rightarrow \quad x = \frac{91}{13} = 7
  \]

#### 4. If \( TV = 14x - 8 \), find \( TU \). Given: \( TU = 9x + 2 \) and \( UV = 5 \).
- Since \( TV = TU + UV \):
  \[
  (9x + 2) + 5 = 14x - 8
  \]
  Simplifying:
  \[
  9x + 7 = 14x - 8 \quad \Rightarrow \quad 7 + 8 = 14x - 9x \quad \Rightarrow \quad 15 = 5x \quad \Rightarrow \quad x = 3
  \]
  Substituting \( x = 3 \) into \( TU \):
  \[
  TU = 9(3) + 2 = 27 + 2 = 29
  \]

#### 5. If \( JL = 5x + 2 \), find \( JL \). Given \( JK = 27 \) and \( KL = 3x - 1 \).
- Since \( JL = JK + KL \):
  \[
  27 + (3x - 1) = 5x + 2
  \]
  Simplifying:
  \[
  26 + 3x = 5x + 2 \quad \Rightarrow \quad 26 - 2 = 5x - 3x \quad \Rightarrow \quad 24 = 2x \quad \Rightarrow \quad x = 12
  \]
  Substituting \( x = 12 \) into \( JL \):
  \[
  JL = 5(12) + 2 = 60 + 2 = 62
  \]

#### 6. If \( CE = 7x + 4 \), find the value of \( x \). Given \( CD = x + 3 \) and \( DE = 8x - 9 \).
- Since \( CE = CD + DE \):
  \[
  (x + 3) + (8x - 9) = 7x + 4
  \]
  Simplifying:
  \[
  9x - 6 = 7x + 4 \quad \Rightarrow \quad 9x - 7x = 4 + 6 \quad \Rightarrow \quad 2x = 10 \quad \Rightarrow \quad x = 5
  \]

Would you like more details on any of these solutions?

### Related Questions:
1. How would you apply the segment addition postulate to three-dimensional geometry?
2. Can two non-collinear points form a congruent segment?
3. How would these rules change in non-Euclidean geometry?
4. What is the importance of congruence in determining shapes' properties?
5. Can you think of a real-life application of measuring and adding segments?

### Tip:
Always double-check your algebra when solving for variables, especially when combining multiple terms or distributing values!

Worksheet on measuring segments, congruent segments, and the segment addition postulate. It includes questions such as finding segment lengths and solving for unknown variables using algebraic expressions.

This worksheet covers geometry problems involving measuring segments, congruent segments, and applying the segment addition postulate. It includes detailed algebraic solutions for segment lengths and unknown variables. Perfect for students in Grades 7-9.

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