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To find the angle \( x \), let's analyze the given geometric figure step by step.

1. **Given Information:**
   - \(\triangle ABC\) is an isosceles triangle with \(AB = AC\).
   - \( \angle ABC = \angle ACB = x \).
   - \( \angle BAC = 75^\circ \).
   - \( D \) is a point on \( BC \) such that \( AD \) bisects \( \angle BAC \).

2. **Isosceles Triangle Property:**
   - In an isosceles triangle, the base angles are equal.
   - Therefore, \(\angle ABC = \angle ACB = x\).

3. **Sum of Angles in \(\triangle ABC\):**
   - The sum of angles in a triangle is \(180^\circ\).
   - \(\angle BAC + \angle ABC + \angle ACB = 180^\circ\).
   - Given that \(\angle BAC = 75^\circ\) and \(\angle ABC = \angle ACB = x\), we have:
   \[
   75^\circ + x + x = 180^\circ
   \]
   \[
   75^\circ + 2x = 180^\circ
   \]
   \[
   2x = 180^\circ - 75^\circ
   \]
   \[
   2x = 105^\circ
   \]
   \[
   x = 52.5^\circ
   \]

Therefore, the angle \( x \) is \( 52.5^\circ \).

Would you like more details or have any other questions?

Here are some questions you might want to ask:
1. Can you explain the properties of isosceles triangles in more detail?
2. How do you determine the sum of angles in a triangle?
3. What are the steps to bisect an angle in a triangle?
4. Can you solve for angles in other types of triangles, like equilateral or scalene?
5. How do you use angle bisectors in solving geometric problems?

**Tip:** Always check the properties of triangles and their internal angles when solving geometric problems.

Learn how to find the angle x in an isosceles triangle with given angles, including detailed steps and application of triangle properties. This problem covers geometry concepts such as triangle properties and angle bisector theorem, suitable for high school students.

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