ABCD is a rectangle with `angle BAC = 48^(@)`. Then `angle DBC` equals

Options

A. `38^(@)`
B. `48^(@)`
C. `132^(@)`
D. `42^(@)`

Answer

To solve the problem, we need to find the measure of angle DBC in rectangle ABCD, given that angle BAC is 48 degrees.

### Step-by-Step Solution:

1. **Understanding the Rectangle**:
In rectangle ABCD, the diagonals AC and BD intersect at point O. Since ABCD is a rectangle, we know that opposite angles are equal, and adjacent angles are supplementary (sum to 90 degrees).

2. **Identifying Angles**:
We are given that angle BAC = 48 degrees. This angle is formed by diagonal AC and side AB.

3. **Using Properties of Diagonals**:
In a rectangle, the diagonals bisect each other. Therefore, angle AOB (where O is the intersection of diagonals) will also be equal to angle COD, and angle AOD will equal angle BOC.

4. **Finding Angle DBC**:
To find angle DBC, we can use the relationship between the angles in triangle BOC.
- Angle DBC is equal to angle CBA minus angle DBA.
- Since ABCD is a rectangle, angle CBA = 90 degrees (as it is one of the angles of the rectangle).

5. **Calculating Angle DBA**:
Since angle BAC = 48 degrees, angle DBA (which is the angle opposite to angle BAC in triangle AOB) will also be equal to angle BAC because the diagonals bisect each other. Therefore, angle DBA = 48 degrees.

6. **Final Calculation**:
Now we can calculate angle DBC:
\[ \text{Angle DBC} = \text{Angle CBA} - \text{Angle DBA} = 90^\circ - 48^\circ = 42^\circ \]

### Conclusion:
Thus, angle DBC equals 42 degrees.