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Let's analyze the given problem:

We are asked to find the measure of \( \angle ACB \) in the provided figure. Here’s what is provided:

- \( \angle A = 60^\circ \)
- \( CE \parallel BA \) (this means \( \overrightarrow{CE} \parallel \overrightarrow{BA} \))
- \( \angle ECD = 65^\circ \)

Since \( CE \parallel BA \), the corresponding angle theorem can be used. This theorem states that when a transversal crosses two parallel lines, the corresponding angles formed are equal.

Thus, since \( CE \parallel BA \), the angle \( \angle ECD \) (which is \( 65^\circ \)) is equal to \( \angle ACB \), because they are corresponding angles.

Hence, \( \angle ACB = 65^\circ \).

Would you like further clarification or details? Here are some related questions for better understanding:

1. What are alternate interior angles, and how are they different from corresponding angles?
2. Can this concept be extended to three-dimensional figures?
3. How would the calculation change if \( CE \) were not parallel to \( BA \)?
4. What is the sum of interior angles in a triangle, and how does that relate to this problem?
5. How do parallel line properties apply in other geometric shapes, like trapezoids?

**Tip**: Always check if lines are parallel, as this often simplifies the use of angle theorems like alternate interior angles or corresponding angles!

In the adjoining figure, it is given that angle A = 60 degrees, CE || BA, and angle ECD = 65 degrees. Find the measure of angle ACB.

This geometry problem asks to find the measure of angle ACB given that angle A = 60 degrees, CE is parallel to BA, and angle ECD = 65 degrees. Learn how to apply the Corresponding Angle Theorem to solve the problem.

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