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The problem asks for the additional information required to determine whether triangle \( \triangle ABC \) is congruent to triangle \( \triangle EDC \). Let's break it down:

### Given Information:
1. \( AC \) and \( BD \) intersect at \( C \).
2. \( AB = ED = 10 \), which means the two triangles have a pair of corresponding sides equal.
3. We need to find the additional piece of information that would make the triangles congruent.

### Triangles and Congruence:
To prove two triangles congruent, we can use the following congruence criteria:
- **SSS (Side-Side-Side)**
- **SAS (Side-Angle-Side)**
- **ASA (Angle-Side-Angle)**
- **AAS (Angle-Angle-Side)**
- **HL (Hypotenuse-Leg) for right triangles**

We already know one pair of sides is equal: \( AB = ED \). To make the triangles congruent, we need more information about either another pair of sides or the angles.

### Analyzing the Choices:
- **A) \( AC = BC \):** This would not necessarily help since it only provides information about triangle \( ABC \), but not triangle \( EDC \). It doesn’t give us information to compare corresponding parts of both triangles.
  
- **B) \( AC = CE \):** This would give us two pairs of equal sides: \( AB = ED \) and \( AC = CE \). If we also know that \( \angle ACB \) and \( \angle EDC \) are equal (which would follow from vertical angles), we could use **SAS (Side-Angle-Side)** to conclude that the triangles are congruent.
  
- **C) The measures of angles \( A \) and \( B \) are equal:** This only gives information about triangle \( ABC \) and doesn't help with triangle \( EDC \), so it wouldn't be sufficient.
  
- **D) The measures of angles \( A \) and \( E \) are equal:** This would help us establish that the triangles have one pair of corresponding angles equal, but without knowing more about sides or other angles, this wouldn’t be enough for congruence.

### Conclusion:
The correct answer is **B) \( AC = CE \)**, because with this information and the given vertical angles at \( C \), the triangles would be congruent by **SAS**.

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Would you like more details or have any questions? Here are five related questions for further exploration:

1. How does the **SAS** postulate work in triangle congruence?
2. What are other methods for proving triangle congruence, besides **SAS**?
3. Can we conclude congruence if only two angles are known to be equal in both triangles?
4. Why does option D not provide sufficient information for congruence?
5. How does knowing vertical angles help in triangle congruence?

**Tip:** When solving triangle congruence problems, always check if the information can be linked using one of the specific congruence criteria (SSS, SAS, ASA, AAS, or HL).

In the figure shown, segments AE and BD intersect at C. Which additional piece of information is sufficient to determine whether triangle ABC is congruent to triangle EDC?

Learn how to determine the congruence of triangles ABC and EDC by applying the SAS (Side-Angle-Side) criterion. This problem also explores the vertical angles theorem and examines other triangle congruence theorems such as SSS and ASA. Suitable for students in Grades 8-10.

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