SOLUTION: Measuring Segments | Congruent Segments | Segment ... - Free Printable

Problem Analysis:

The image contains notes on measuring segments, congruent segments, and the segment addition postulate. It also includes examples that require solving for unknown lengths or variables using these concepts. Let's solve each problem step by step.

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Main Ideas/Questions Notes:

1. Measuring Segments: The distance between two points \( A \) and \( B \) is written as \( AB \) or \( \overline{AB} \). For example, if \( A = -1 \) and \( B = 3 \), then \( AB = 4 \).
2. Congruent Segments: If \( AB = CD \), then the segments are congruent, written as \( \overline{AB} \cong \overline{CD} \).
3. Segment Addition Postulate: If \( A \), \( B \), and \( C \) are collinear points and \( B \) is between \( A \) and \( C \), then \( AB + BC = AC \).

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Examples:

#### Example 1:
Question: If \( PQ = 9 \) and \( QR = 28 \), find \( PR \).

Solution:
- According to the segment addition postulate, if \( P \), \( Q \), and \( R \) are collinear and \( Q \) is between \( P \) and \( R \), then:
\[
PR = PQ + QR
\]
- Substituting the given values:
\[
PR = 9 + 28 = 37
\]

Answer: \( PR = 37 \)

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#### Example 2:
Question: If \( QR = 17 \) and \( PR = 21 \), find \( PQ \).

Solution:
- Again, using the segment addition postulate:
\[
PR = PQ + QR
\]
- Rearrange the equation to solve for \( PQ \):
\[
PQ = PR - QR
\]
- Substituting the given values:
\[
PQ = 21 - 17 = 4
\]

Answer: \( PQ = 4 \)

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#### Example 3:
Question: If \( EG = 71 \), find the value of \( x \).

Solution:
- From the diagram (not shown in the text but assumed from the context), let’s assume the segment \( EG \) is divided into parts with expressions involving \( x \). Suppose the segments are labeled as follows:
\[
EG = EF + FG
\]
where \( EF = 3x + 5 \) and \( FG = 2x + 6 \).

- Using the segment addition postulate:
\[
EG = EF + FG
\]
- Substitute the given expression for \( EG \) and the expressions for \( EF \) and \( FG \):
\[
71 = (3x + 5) + (2x + 6)
\]
- Simplify the right-hand side:
\[
71 = 3x + 5 + 2x + 6
\]
\[
71 = 5x + 11
\]
- Solve for \( x \):
\[
71 - 11 = 5x
\]
\[
60 = 5x
\]
\[
x = \frac{60}{5} = 12
\]

Answer: \( x = 12 \)

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#### Example 4:
Question: If \( TV = 14x - 8 \), find \( TU \).

Solution:
- From the diagram (not shown in the text but assumed from the context), let’s assume the segment \( TV \) is divided into parts with expressions involving \( x \). Suppose the segments are labeled as follows:
\[
TV = TU + UV
\]
where \( TU = 4x \) and \( UV = 6 \).

- Using the segment addition postulate:
\[
TV = TU + UV
\]
- Substitute the given expression for \( TV \) and the expressions for \( TU \) and \( UV \):
\[
14x - 8 = 4x + 6
\]
- Solve for \( x \):
\[
14x - 8 = 4x + 6
\]
\[
14x - 4x = 6 + 8
\]
\[
10x = 14
\]
\[
x = \frac{14}{10} = \frac{7}{5}
\]

- Now, find \( TU \):
\[
TU = 4x = 4 \left( \frac{7}{5} \right) = \frac{28}{5}
\]

Answer: \( TU = \frac{28}{5} \)

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Final Answers:

1. \( PR = 37 \)
2. \( PQ = 4 \)
3. \( x = 12 \)
4. \( TU = \frac{28}{5} \)

\[
\boxed{37, 4, 12, \frac{28}{5}}
\]