If AP and BP are the bisectors of the angle A and angle B of a parallelogram ABCD, then value of the angle APB is

Options

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

Answer

To find the value of angle APB in the given parallelogram ABCD, where AP and BP are the angle bisectors of angles A and B respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Properties of a Parallelogram**:
- In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. Therefore, if angle A is denoted as ∠A and angle B as ∠B, we have:
\[ ∠A + ∠B = 180^\circ \]

2. **Using the Angle Bisector Theorem**:
- Since AP is the bisector of angle A, we can express angle A as:
\[ ∠A = 2 \times ∠1 \]
where ∠1 is half of angle A.
- Similarly, since BP is the bisector of angle B, we can express angle B as:
\[ ∠B = 2 \times ∠2 \]
where ∠2 is half of angle B.

3. **Setting Up the Equation**:
- From the property of the parallelogram, we know:
\[ ∠A + ∠B = 180^\circ \implies 2∠1 + 2∠2 = 180^\circ \]
- Dividing the entire equation by 2 gives:
\[ ∠1 + ∠2 = 90^\circ \quad \text{(Equation 1)} \]

4. **Analyzing Triangle APB**:
- In triangle APB, we apply the angle sum property, which states that the sum of the angles in a triangle is 180 degrees:
\[ ∠1 + ∠2 + ∠APB = 180^\circ \]

5. **Substituting Equation 1 into the Triangle Equation**:
- We already established that ∠1 + ∠2 = 90° from Equation 1. Substituting this into the triangle equation gives:
\[ 90^\circ + ∠APB = 180^\circ \]

6. **Solving for Angle APB**:
- To find ∠APB, we rearrange the equation:
\[ ∠APB = 180^\circ - 90^\circ = 90^\circ \]

### Conclusion:
Thus, the value of angle APB is:
\[ \boxed{90^\circ} \]