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The problem you've shared involves a quadrilateral where certain angles and sides are given, and the question specifically asks for the value of angle \( \angle ADB \).

### Given Information:
- \( \angle ABC = 150^\circ \)
- \( \angle BCD = 30^\circ \)
- \( AB = BD = BC \) (the sides marked as equal in the diagram)

#### To Find: \( \angle ADB \)

### Step-by-Step Solution:
1. **Symmetry of the triangle**: 
   Since \( AB = BD = BC \), triangle \( \triangle ABD \) is isosceles. This symmetry gives us useful relationships between the angles.
   
2. **Use the sum of angles in triangle \( \triangle ABC \)**:
   In triangle \( ABC \), we know:
   \[
   \angle ABC = 150^\circ \quad \text{and} \quad \angle BCD = 30^\circ.
   \]
   Now, since \( AB = BC \), the triangle \( \triangle ABC \) is isosceles, and thus the base angles (at \( A \) and \( C \)) are equal.

3. **Find the unknown angle in triangle \( \triangle ABC \)**:
   The sum of angles in a triangle is always \( 180^\circ \), so:
   \[
   \angle BAC + \angle BCA + \angle ABC = 180^\circ.
   \]
   Substituting the given value \( \angle ABC = 150^\circ \):
   \[
   \angle BAC + \angle BCA + 150^\circ = 180^\circ.
   \]
   This simplifies to:
   \[
   \angle BAC + \angle BCA = 30^\circ.
   \]
   Since the two base angles are equal:
   \[
   \angle BAC = \angle BCA = 15^\circ.
   \]

4. **Determine \( \angle ADB \)**:
   Since triangle \( ABD \) is isosceles (with \( AB = BD \)), the angles at \( A \) and \( D \) are equal. Therefore, in triangle \( ABD \), we can use the fact that the total sum of angles is \( 180^\circ \):
   \[
   2 \times \angle ADB + \angle ABD = 180^\circ.
   \]
   But \( \angle ABD = 30^\circ \), so:
   \[
   2 \times \angle ADB + 30^\circ = 180^\circ,
   \]
   which simplifies to:
   \[
   2 \times \angle ADB = 150^\circ,
   \]
   giving:
   \[
   \angle ADB = 75^\circ.
   \]

Thus, the measure of \( \angle ADB \) is \( 75^\circ \).

---

Would you like any further details or clarifications on this solution?

Here are 5 related questions to explore further:
1. How would the problem change if angle \( \angle ABC \) were different?
2. Can you prove the properties of isosceles triangles using coordinate geometry?
3. What would happen to the angles if the sides were not equal (i.e., \( AB \neq BD \neq BC \))?
4. How would you find the area of this quadrilateral?
5. What are some real-world applications of isosceles triangles?

**Tip**: When dealing with isosceles triangles, always try to use symmetry to simplify the problem and reduce unknown variables.

Find the value of angle ADB in a quadrilateral where AB = BD = BC, angle ABC = 150°, and angle BCD = 30°.

This problem involves finding the value of angle ADB in a quadrilateral with given angles and side relationships. We use isosceles triangle properties and the angle sum theorem to solve the problem.

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