https://file.aiquickdraw.com/mathbot/file/1726171386336gw2a85wx.jpg

To solve for \( x \), let's analyze the given geometry:

1. **Identify the angles and relationships:**
   - We have a line segment \( BC \) divided at point \( D \).
   - The triangles \( \triangle ABD \) and \( \triangle ACD \) share side \( AD \).
   - Angles \( \angle BDA \) and \( \angle ADC \) are equal because the opposite sides are marked as equal, indicating congruence by symmetry.

2. **Use the given angle measures:**
   - \( \angle BDA = 45^\circ \)
   - \( \angle ACB = 30^\circ \)

3. **Sum of angles:**
   - In triangle \( \triangle ABD \):
     \[
     \angle ABD = 180^\circ - (x + 45^\circ) = 135^\circ - x
     \]
   - In triangle \( \triangle ACD \):
     \[
     \angle CAD = 180^\circ - (x + 30^\circ) = 150^\circ - x
     \]

4. **Isosceles triangle properties:**
   - Since \( \triangle ACD \) and \( \triangle ABD \) share properties (isosceles with equal segments), \( \angle ABD = \angle ACD = x \).

5. **Set up equations using angle sums:**
   - For \( \triangle ABD \) and \( \triangle ACD \):
     \[
     x + 45^\circ + x = 180^\circ \implies 2x + 45^\circ = 180^\circ \implies 2x = 135^\circ \implies x = 67.5^\circ
     \]

Therefore, the value of \( x \) is \( 67.5^\circ \).

Do you want details on this explanation or have any questions?

Here are some related questions for further practice:
1. What are the properties of isosceles triangles?
2. How do angle sum properties apply in different triangles?
3. What methods can be used to prove triangles are congruent?
4. How can external angles help in finding internal angles?
5. What are the rules of supplementary and complementary angles?

**Tip:** Always check for symmetrical properties in geometric figures to simplify calculations!

Find the value of x in the given geometric configuration where angle measures and congruent segments are indicated.

Learn how to solve for x in a geometric triangle by applying the angle sum theorem and properties of isosceles triangles. This problem is suitable for high school students and involves the use of congruent triangle properties and basic geometry rules.

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