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We are given that \( WZ = 128 \, \text{ft} \), and \( WX \cong YZ \), which means that \( WX \) and \( YZ \) are congruent, or equal in length.

Let's break down the problem step-by-step:

1. Since \( WX \cong YZ \), we have \( WX = YZ \). Let’s call the length of \( WX \) (and thus \( YZ \)) as \( x \).
2. From the diagram, we also know that \( XY = 54 \, \text{ft} \).

Now, since \( WZ = WX + XY + YZ \), we can express this as:
\[
WZ = x + 54 + x
\]
Substitute \( WZ = 128 \, \text{ft} \):
\[
128 = x + 54 + x
\]
Simplify the equation:
\[
128 = 2x + 54
\]
Solve for \( x \):
\[
2x = 128 - 54
\]
\[
2x = 74
\]
\[
x = 37
\]

Thus, \( WX = 37 \, \text{ft} \) and \( YZ = 37 \, \text{ft} \).

Now, to find \( WY \), we add \( WX \) and \( XY \):
\[
WY = WX + XY = 37 + 54 = 91 \, \text{ft}
\]

Therefore, the length of \( WY \) is \( 91 \, \text{ft} \).

Would you like a further breakdown of the steps? 

Here are 5 questions to expand on this:

1. What is the reasoning behind \( WX \cong YZ \) leading to equal lengths for those segments?
2. How would the solution change if \( WZ \) were a different total length?
3. Can we generalize this method to any similar problem with congruent segments?
4. What role does segment addition play in solving problems like these?
5. How can we check if our solution is correct by working backwards?

**Tip**: When dealing with congruent segments, always look for ways to express them as equal variables—it simplifies the process!

Find the length of WY if WZ = 128 ft and WX ≅ YZ.

Solve a geometry problem involving congruent segments and find the length of WY. Given that WZ = 128 ft and WX ≅ YZ, this step-by-step solution applies congruence and the segment addition postulate. Suitable for students in Grades 7-9.

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