To solve this problem, we need to use similar triangles and the properties of a triangle's perimeter. Let's analyze the given triangle $\triangle ABC$ and the smaller triangle $\triangle ADE$ that is formed by the line segment $DE$.

Step-by-Step Solution:

1. Given information:

   • $AB = 30$ in.
   • $AC = 50$ in.
   • $BC = 60$ in.
   • $P_{ADE} = AD + DE + AE = 56$ in.
   • $\angle AED = \angle ABC$ implies that $\triangle ADE \sim \triangle ABC$ (angle-angle similarity).

2. Determine the similarity ratio: Since $\triangle ADE \sim \triangle ABC$, the sides of $\triangle ADE$ are proportional to the sides of $\triangle ABC$.

   Let's denote the similarity ratio by $k$. Thus, the sides of $\triangle ADE$ can be expressed as:

   • $AD = k \cdot AB$
   • $AE = k \cdot AC$
   • $DE = k \cdot BC$

3. Express the perimeter of $\triangle ADE$ in terms of $k$: The perimeter of $\triangle ADE$ is given by: $AD + DE + AE = k \cdot AB + k \cdot BC + k \cdot AC$

   Substituting the known lengths: $k \cdot 30 + k \cdot 60 + k \cdot 50 = 56$

   Simplify this expression: $k (30 + 60 + 50) = 56$ $k \cdot 140 = 56$

   Solve for $k$: $k = \frac{56}{140} = \frac{2}{5}$

4. Find $AD$, $DE$, and $AE$: Using $k = \frac{2}{5}$:

   • $AD = k \cdot AB = \frac{2}{5} \times 30 = 12$ in.
   • $AE = k \cdot AC = \frac{2}{5} \times 50 = 20$ in.
   • $DE = k \cdot BC = \frac{2}{5} \times 60 = 24$ in.

5. Calculate $BD$ and $CE$: To find $BD$ and $CE$, we calculate the remaining parts of sides $AB$ and $AC$, respectively.

   • $BD = AB - AD = 30 - 12 = 18$ in.
   • $CE = AC - AE = 50 - 20 = 30$ in.

6. Sum of $BD$ and $CE$: $BD + CE = 18 + 30 = 48 \text{ in.}$

Conclusion:

The sum of the lengths of line segments $BD$ and $CE$ is 48 inches.

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Would you like to ask for details or have any questions?

Here are five related questions for further exploration:

1. How do you determine the similarity ratio between two similar triangles?
2. What is the significance of angle similarity in triangles?
3. Can you find the area ratio of $\triangle ADE$ to $\triangle ABC$?
4. What would be the length of $BE$ in this problem?
5. How can you use the similarity ratio to find the lengths of all segments of a triangle?

Tip: Always check if triangles are similar using the angle-angle (AA) criterion, as it can simplify finding unknown lengths.