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We are given that \( ABCD \) is a rhombus and \( \angle DBC = 55^\circ \). We need to find \( \angle DCA \).

### Key Properties of Rhombus:
1. All sides of a rhombus are equal.
2. The diagonals of a rhombus bisect each other at right angles (90°).
3. The diagonals bisect the angles of the rhombus.

#### Step-by-Step Solution:
1. Since \( ABCD \) is a rhombus, the diagonals bisect each other at right angles, and they bisect the angles at the vertices.
   
2. We know that \( \angle DBC = 55^\circ \). This angle is part of \( \triangle DBC \), and since \( \triangle DBC \) is an isosceles triangle (because sides \( BD = BC \), being sides of the rhombus), the angle \( \angle DCB \) will also be \( 55^\circ \).

3. Now, in \( \triangle DBC \), the sum of the angles must be \( 180^\circ \). So, we can find \( \angle BDC \) as follows:
   \[
   \angle BDC = 180^\circ - \angle DBC - \angle DCB = 180^\circ - 55^\circ - 55^\circ = 70^\circ.
   \]

4. The diagonals of a rhombus bisect the opposite angles, so \( \angle DCA \) is half of \( \angle BDC \). Therefore:
   \[
   \angle DCA = \frac{1}{2} \times \angle BDC = \frac{1}{2} \times 70^\circ = 35^\circ.
   \]

Thus, \( \angle DCA = 35^\circ \).

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:
1. What are the properties of a rhombus that distinguish it from other quadrilaterals?
2. How do the diagonals of a rhombus affect its internal angles?
3. Can you prove that the diagonals of a rhombus bisect the angles?
4. If \( \angle DBC \) were changed, how would that affect \( \angle DCA \)?
5. How would the solution change if the rhombus were instead a square?

**Tip:** In rhombuses, the diagonals always divide the rhombus into four right-angled triangles.

In the given figure, ABCD is a rhombus. If ∠DBC = 55°, find ∠DCA.

Learn how to find ∠DCA in a rhombus ABCD where ∠DBC = 55°. This solution uses properties of rhombus and angle sum theorem to solve the problem, suitable for students in Grades 9-10.

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