Worksheet for constructing congruent segments and performing geometric operations with line segments.

Problem Overview:

The worksheet focuses on constructing congruent segments using geometric tools such as a straightedge and compass. The goal is to replicate given segments or combine them in specific ways (e.g., addition or subtraction of lengths).

Solution Explanation:

#### General Steps for Constructing a Congruent Segment:
1. Draw a Working Line: Use a straightedge to draw a line where the new segment will be constructed.
2. Choose a Starting Point: Select a point on the working line and label it (e.g., \( A' \)).
3. Set the Compass Radius: Place the compass at one endpoint of the given segment (e.g., \( A \)) and adjust it so that the pencil tip is at the other endpoint (e.g., \( B \)). This sets the radius equal to the length of the given segment.
4. Transfer the Length: Place the compass point at the chosen starting point on the working line (e.g., \( A' \)) and draw an arc. The intersection of this arc with the working line gives the endpoint of the new segment (e.g., \( B' \)).
5. Label the New Segment: The segment \( A'B' \) is now congruent to the original segment \( AB \).

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Detailed Solutions for Each Problem:

#### 1. Construct a segment congruent to \( \overline{CD} \):
- Step 1: Draw a working line.
- Step 2: Choose a point on the working line and label it \( C' \).
- Step 3: Set the compass radius equal to the length of \( \overline{CD} \) by placing the compass at \( C \) and adjusting it to \( D \).
- Step 4: Place the compass point at \( C' \) and draw an arc intersecting the working line. Label the intersection point \( D' \).
- Step 5: The segment \( \overline{C'D'} \) is congruent to \( \overline{CD} \).

#### 2. Construct a segment congruent to \( \overline{EF} \):
- Follow the same steps as above, but use \( \overline{EF} \) as the reference segment. Label the new segment \( \overline{E'F'} \).

#### 3. Construct a segment congruent to \( \overline{XY} \):
- Again, follow the same steps, using \( \overline{XY} \) as the reference segment. Label the new segment \( \overline{X'Y'} \).

#### 4. Construct a segment whose length is \( \overline{CD} + \overline{EF} \):
- Step 1: Draw a working line.
- Step 2: Choose a point on the working line and label it \( P \).
- Step 3: Construct a segment \( \overline{PQ} \) congruent to \( \overline{CD} \) starting from \( P \).
- Step 4: Without changing the compass setting, place the compass point at \( Q \) and construct a segment \( \overline{QR} \) congruent to \( \overline{EF} \).
- Step 5: The segment \( \overline{PR} \) is the desired segment, with length \( \overline{CD} + \overline{EF} \).

#### 5. Construct a segment whose length is \( \overline{EF} + \overline{XY} \):
- Follow the same steps as in Problem 4, but use \( \overline{EF} \) and \( \overline{XY} \) as the reference segments. The resulting segment will be \( \overline{EF} + \overline{XY} \).

#### 6. Construct a segment whose length is \( \overline{EF} - \overline{CD} \):
- Step 1: Draw a working line.
- Step 2: Choose a point on the working line and label it \( S \).
- Step 3: Construct a segment \( \overline{ST} \) congruent to \( \overline{EF} \).
- Step 4: Without changing the compass setting, place the compass point at \( S \) and mark an arc intersecting \( \overline{ST} \) at a point \( U \) such that \( \overline{SU} \) is congruent to \( \overline{CD} \).
- Step 5: The segment \( \overline{UT} \) is the desired segment, with length \( \overline{EF} - \overline{CD} \).

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Final Answer:

\[
\boxed{
\text{Follow the steps outlined above for each construction.}
}
\]