ABCD is a rhombus. If `angleACB=30^(@)`, then `angleADB` is

Options

A. `30^(@)`
B. `120^(@)`
C. `60^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to find the angle \( \angle ADB \) in the rhombus \( ABCD \) given that \( \angle ACB = 30^\circ \).

### Step-by-Step Solution:

1. **Understanding the Rhombus Properties**:
- In a rhombus, opposite angles are equal, and adjacent angles are supplementary (i.e., they add up to \( 180^\circ \)).
- The diagonals of a rhombus bisect each other at right angles.

2. **Identifying Angles**:
- We are given \( \angle ACB = 30^\circ \).
- Since \( ABCD \) is a rhombus, \( AC \) and \( BD \) are the diagonals.

3. **Finding \( \angle ACD \)**:
- Since \( AC \) and \( BD \) bisect each other at right angles, \( \angle ACD \) will also be \( 30^\circ \) because \( \angle ACB \) and \( \angle ACD \) are alternate interior angles formed by the transversal \( AC \) with the parallel lines \( AB \) and \( CD \).

4. **Using the Triangle Sum Property**:
- In triangle \( ACD \), we know:
\[ \angle DAC + \angle ACD + \angle ADB = 180^\circ \]
- We know \( \angle ACD = 30^\circ \) and \( \angle DAC = 90^\circ \) (since diagonals bisect each other at right angles).

5. **Calculating \( \angle ADB \)**:
- Substitute the known angles into the triangle sum equation:
\[ 90^\circ + 30^\circ + \angle ADB = 180^\circ \]
- Simplifying gives:
\[ \angle ADB = 180^\circ - 120^\circ = 60^\circ \]

### Final Answer:
Thus, \( \angle ADB = 60^\circ \).

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