Illustration of congruent line segments and a number line demonstrating segment congruence.

Problem Analysis:

The task involves analyzing the given geometric relationships and determining the lengths of segments based on the provided congruence conditions. Let's break it down step by step.

#### Given Information:
1. Segment Congruences:
- \( AB \cong CD \)
- \( EF \cong GH \)
- \( KL \cong MN \)

2. Explicit Lengths:
- \( EF = 5 \, \text{cm} \)
- \( GH = 5 \, \text{cm} \)

3. Coordinate System:
- The coordinates of points \( K, L, M, N \) are given on a number line:
- \( K = -4 \)
- \( L = -2 \)
- \( M = 1 \)
- \( N = 3 \)

#### Objective:
Determine the lengths of the segments \( AB \), \( CD \), \( KL \), and \( MN \) using the given congruence relationships and coordinate information.

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Step-by-Step Solution:

#### 1. Analyzing \( EF \cong GH \):
- It is explicitly given that \( EF = 5 \, \text{cm} \).
- Since \( EF \cong GH \), the length of \( GH \) must also be \( 5 \, \text{cm} \).

Therefore:
\[
EF = GH = 5 \, \text{cm}
\]

#### 2. Analyzing \( AB \cong CD \):
- The segments \( AB \) and \( CD \) are congruent, meaning they have the same length.
- However, the problem does not provide explicit lengths for \( AB \) or \( CD \). To determine their lengths, we need additional information or assumptions. Since no further details are given, we can only state that:
\[
AB = CD
\]
The exact numerical value cannot be determined without additional data.

#### 3. Analyzing \( KL \cong MN \):
- The segments \( KL \) and \( MN \) are congruent, meaning they have the same length.
- We can calculate the lengths of \( KL \) and \( MN \) using the coordinates provided on the number line.

- Length of \( KL \):
\[
KL = |L - K| = |-2 - (-4)| = |-2 + 4| = 2 \, \text{cm}
\]

- Length of \( MN \):
\[
MN = |N - M| = |3 - 1| = 2 \, \text{cm}
\]

Since \( KL \cong MN \), both segments have the same length:
\[
KL = MN = 2 \, \text{cm}
\]

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Final Answer:

Summarizing the lengths of all segments:
\[
\boxed{AB = CD, \, EF = GH = 5 \, \text{cm}, \, KL = MN = 2 \, \text{cm}}
\]

Note: The exact numerical value of \( AB \) and \( CD \) cannot be determined without additional information.