A diagonal of a rectangle is inclined to one side of a rectangle at `34^(@)`. The acute angle between the diagonals is :

Options

A. `34^(@)`
B. `56^(@)`
C. `68^(@)`
D. `42^(@)`

Answer

To find the acute angle between the diagonals of a rectangle when one diagonal is inclined to one side of the rectangle at \(34^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Rectangle and Diagonals**:
- Let the rectangle be \(ABCD\) with vertices \(A\), \(B\), \(C\), and \(D\).
- The diagonals \(AC\) and \(BD\) intersect at point \(O\).

2. **Identifying the Given Angle**:
- The diagonal \(AC\) is inclined to side \(AD\) at an angle of \(34^\circ\).
- This means that \(\angle ODC = 34^\circ\).

3. **Finding Angle ADO**:
- Since the angles in a rectangle are \(90^\circ\), we can find \(\angle ADO\) using:
\[ \angle ADO = 90^\circ - \angle ODC \]
- Substituting the value:
\[ \angle ADO = 90^\circ - 34^\circ = 56^\circ \]

4. **Using Properties of Diagonals**:
- The diagonals of a rectangle bisect each other and are equal in length.
- Therefore, \(\angle OAD\) will also be equal to \(\angle ADO\):
\[ \angle OAD = 56^\circ \]

5. **Finding the Angle AOD**:
- Now, we know that the sum of angles in triangle \(AOD\) is \(180^\circ\):
\[ \angle AOD + \angle OAD + \angle ADO = 180^\circ \]
- Substituting the known values:
\[ \angle AOD + 56^\circ + 56^\circ = 180^\circ \]
- Simplifying:
\[ \angle AOD + 112^\circ = 180^\circ \]
- Therefore:
\[ \angle AOD = 180^\circ - 112^\circ = 68^\circ \]

6. **Identifying the Acute Angle**:
- The acute angle between the diagonals \(AC\) and \(BD\) is \(\angle AOD\), which we calculated to be \(68^\circ\).

### Final Answer:
The acute angle between the diagonals of the rectangle is \(68^\circ\).

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