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The image you've uploaded shows a geometric problem involving a kite, where you need to find angles within the kite formed by isosceles triangles. Let's analyze the situation and find the necessary angle relationships.

### Explanation:

1. **Kite Properties**:
   - A kite is a quadrilateral with two pairs of adjacent sides that are equal.
   - The diagonals of a kite intersect at right angles (90 degrees).

2. **Given Angles**:
   - The angle at vertex \(A\) is \(2x\).
   - The angle at vertex \(B\) is \(3x\).
   - The angles at vertices \(C\) and \(D\) are both \(x\).

### Step-by-Step Solution:

1. **Using the properties of a kite:**
   - The diagonals of a kite are perpendicular, meaning the angle between the diagonals is \(90^\circ\).
   - In any quadrilateral, the sum of the interior angles is \(360^\circ\).

2. **Sum of Angles in Triangle**:
   - In \( \triangle ABC \), using the sum of the angles:
     \[
     \angle A + \angle B + \angle C = 180^\circ
     \]
   - Substitute the given angles:
     \[
     2x + 3x + x = 180^\circ
     \]
     \[
     6x = 180^\circ
     \]
     \[
     x = 30^\circ
     \]

3. **Substituting \( x \)**:
   - \( \angle A = 2x = 2(30^\circ) = 60^\circ \)
   - \( \angle B = 3x = 3(30^\circ) = 90^\circ \)
   - \( \angle C = \angle D = 30^\circ \)

### Conclusion:
- The angles at vertices \(A\) and \(B\) are \(60^\circ\) and \(90^\circ\), respectively, and at \(C\) and \(D\), both are \(30^\circ\). The diagonal intersection angle is \(90^\circ\) as expected for a kite.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How do you prove that the diagonals of a kite are perpendicular?
2. What is the relationship between the sides of a kite when the diagonals intersect?
3. How would the angle calculations change if the kite were not symmetric?
4. What other quadrilateral properties can be applied to solve similar problems?
5. How do you calculate the area of a kite using its diagonals?

**Tip**: Remember that the sum of angles in any quadrilateral is always \(360^\circ\).

How do you find the angle made by two isosceles triangles in a kite?

Learn how to calculate the angles in a kite formed by isosceles triangles with a step-by-step explanation. Explore key geometric principles such as the sum of angles in triangles and quadrilaterals. Suitable for students in Grades 8-10.

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