To solve for $BD$, let’s break down the information given in the image.

1. Congruent Triangles: The problem states that $\triangle ABC$ is congruent to $\triangle DBC$.

2. Side Lengths:

   • $AB = AC = 4$ inches (given).
   • $BC$ is a common side for both triangles.
   • $AD = 5$ inches (given as the length connecting $A$ and $D$).

3. Interpretation of BD: Since $\triangle ABC$ is congruent to $\triangle DBC$, the line $BD$ should be equal to $CD$, splitting $AD$ into two equal segments.

Solution:

Since $AD = 5$ inches, and $BD$ and $CD$ split $AD$ equally due to congruence: $BD = \frac{AD}{2} = \frac{5}{2} = 2.5 \text{ inches}$

Answer:

$BD = 2.5 \text{ inches}$

Would you like more details or have any questions?

Here are some related questions for further practice:

1. If $\triangle ABC$ and $\triangle DBC$ are congruent, what can we say about the angles in both triangles?
2. How would you calculate the perimeter of $\triangle ABC$?
3. What is the area of $\triangle ABC$ if it’s a right triangle?
4. Can you prove congruence using a specific triangle congruence theorem here?
5. How does the concept of symmetry apply in this diagram?

Tip: When triangles are congruent, corresponding sides and angles are equal. This can simplify problems by dividing measurements evenly.