To solve this problem, we need to use the fact that $\triangle CDE$ is isosceles, meaning two sides of the triangle are equal. Specifically, we are given:

1. $CD = 9x - 25$
2. $DE = 6x + 8$
3. $CE = 10x - 73$

Since $CD \cong DE$ (they are equal), we set $CD = DE$ and solve for $x$:

$9x - 25 = 6x + 8$

Step 1: Solve for $x$

Subtract $6x$ from both sides: $3x - 25 = 8$

Add 25 to both sides: $3x = 33$

Divide by 3: $x = 11$

Step 2: Find the measures of each side

Substitute $x = 11$ into the expressions for $CD$, $DE$, and $CE$:

1. $CD = 9x - 25 = 9(11) - 25 = 99 - 25 = 74$
2. $DE = 6x + 8 = 6(11) + 8 = 66 + 8 = 74$
3. $CE = 10x - 73 = 10(11) - 73 = 110 - 73 = 37$

Final Answer:

• $CD = 74$
• $DE = 74$
• $CE = 37$

Let me know if you'd like further clarification or additional steps explained.

Here are 5 questions for deeper exploration:

1. What property of isosceles triangles allows us to solve this problem?
2. Why must $CD$ and $DE$ be equal in this case?
3. How do the side lengths affect the classification of $\triangle CDE$?
4. Could this triangle form a right triangle? Why or why not?
5. How would the solution change if $CE = CD$ instead?

Tip: Always check the triangle inequality theorem to confirm that the given side lengths form a valid triangle.