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The diagram provided shows a geometric figure involving triangles, lines, and angles. The given information states:

- \( BD = DC \)
- \( AB = ED \)
- \( AC = EC \)
- \( AB \parallel ED \)

The specific question asks: "Calculate the angle size of \( \hat{C_1} \) with reasons."

### Analyzing the diagram:

- Since \( BD = DC \), \( BCD \) is an isosceles triangle, meaning angles \( \hat{B_1} \) and \( \hat{D_1} \) at the base are equal.
- \( AB \parallel ED \) suggests that angle relationships could involve corresponding angles or alternate interior angles due to the parallel lines.

### Calculation of \( \hat{C_1} \):

We can use angle properties of triangles and parallel lines to find the measure of \( \hat{C_1} \), depending on the relationships between the angles provided by the parallel lines and the isosceles triangle.

Would you like to proceed with detailed steps on calculating \( \hat{C_1} \), or do you have any specific clarifications about this problem? 

Here are some related questions to explore:

1. What are the properties of isosceles triangles that can help in angle calculations?
2. How do parallel lines influence the angles formed by a transversal?
3. How can alternate interior angles assist in solving this problem?
4. Can the sum of angles in a triangle provide insights into solving for \( \hat{C_1} \)?
5. Are there any symmetry properties in the diagram that simplify the calculations?

**Tip:** When working with isosceles triangles, remember that the base angles are always equal.

Calculate the angle size of \( \hat{C_1} \) with reasons.

Learn how to calculate the angle size of \( \hat{C_1} \) using properties of isosceles triangles and parallel lines. Explore how angle relationships and theorems are applied to solve this geometry problem, suitable for students in Grades 8-10.

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