ABCD is a rhombus such that `angleACB=40^(@)`, then `angleADB` is

Options

A. `40^(@)`
B. `45^(@)`
C. `50^(@)`
D. `60^(@)`

Explanation

To solve the problem step by step, we will find the measure of angle ADB in the rhombus ABCD given that angle ACB is 40 degrees.

### Step-by-Step Solution:

1. **Understanding the Properties of a Rhombus**:
- In a rhombus, the diagonals bisect each other at right angles (90 degrees).
- Therefore, if we denote the intersection point of the diagonals as O, then angle AOD = 90 degrees.

**Hint**: Remember that the diagonals of a rhombus intersect at right angles.

2. **Identifying Alternate Interior Angles**:
- Since AB is parallel to CD and AC is a transversal, angle ACB (which is given as 40 degrees) is equal to angle DAO (alternate interior angles).
- Thus, we can write: angle DAO = angle ACB = 40 degrees.

**Hint**: Use the property of alternate interior angles when two parallel lines are cut by a transversal.

3. **Using the Angle Sum Property in Triangle AOD**:
- In triangle AOD, the sum of the angles is 180 degrees. Therefore, we can write:
\[ \text{angle DAO} + \text{angle AOD} + \text{angle ADB} = 180^\circ \]
- Substituting the known values, we have:
\[ 40^\circ + 90^\circ + \text{angle ADB} = 180^\circ \]

**Hint**: Recall that the sum of angles in any triangle is always 180 degrees.

4. **Solving for angle ADB**:
- Simplifying the equation:
\[ 130^\circ + \text{angle ADB} = 180^\circ \]
- Now, subtract 130 degrees from both sides:
\[ \text{angle ADB} = 180^\circ - 130^\circ = 50^\circ \]

**Hint**: Perform basic arithmetic to isolate the variable you are solving for.

5. **Conclusion**:
- Therefore, the measure of angle ADB is 50 degrees.

**Final Answer**: angle ADB = 50 degrees.