The bisectors of any two adjacent angles of a parallelogram intersect at

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `90^@`

Answer

To solve the question regarding the intersection of the bisectors of any two adjacent angles of a parallelogram, let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Angles in a Parallelogram**:
In a parallelogram, adjacent angles are supplementary, meaning their sum is 180 degrees. Let’s denote the adjacent angles as \( \angle A \) and \( \angle B \). Therefore, we have:
\[ \angle A + \angle B = 180^\circ \]

2. **Using the Angle Bisector**:
The bisector of an angle divides it into two equal parts. Therefore, the angle bisector of \( \angle A \) will create two angles:
\[ \angle OAB = \frac{1}{2} \angle A \]
Similarly, the angle bisector of \( \angle B \) will create:
\[ \angle OBA = \frac{1}{2} \angle B \]

3. **Setting Up the Triangle**:
Now, consider the triangle formed by the intersection of the two bisectors, which we can denote as triangle OAB. The sum of the angles in a triangle is always 180 degrees. Therefore, we can write:
\[ \angle OAB + \angle OBA + \angle AOB = 180^\circ \]

4. **Substituting the Bisector Angles**:
We can substitute the values of \( \angle OAB \) and \( \angle OBA \) into the triangle angle sum equation:
\[ \frac{1}{2} \angle A + \frac{1}{2} \angle B + \angle AOB = 180^\circ \]

5. **Factoring Out the Common Term**:
Since \( \angle A + \angle B = 180^\circ \), we can replace \( \angle A + \angle B \) in our equation:
\[ \frac{1}{2} (180^\circ) + \angle AOB = 180^\circ \]
Simplifying this gives:
\[ 90^\circ + \angle AOB = 180^\circ \]

6. **Solving for \( \angle AOB \)**:
Now, we can isolate \( \angle AOB \):
\[ \angle AOB = 180^\circ - 90^\circ = 90^\circ \]

### Conclusion:
The bisectors of any two adjacent angles of a parallelogram intersect at an angle of \( 90^\circ \). Therefore, the correct option is:
**Option 4: 90 degrees.**