The bisectors of angles of a parallelogram forms a

Options

A. Trapezium
B. Rectangle
C. Rhombus
D. Kite

Answer

To solve the problem of determining what shape is formed by the angle bisectors of a parallelogram, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of a Parallelogram**:
- A parallelogram has opposite angles that are equal and consecutive angles that are supplementary.
- Let’s denote the angles of the parallelogram ABCD as follows: ∠A = 2x, ∠B = 2y, ∠C = 2x, and ∠D = 2y.

**Hint**: Remember that in a parallelogram, opposite angles are equal and the sum of adjacent angles is 180°.

2. **Draw the Angle Bisectors**:
- Draw the angle bisectors of angles A, B, C, and D. The angle bisector of ∠A will divide it into two angles of x each. Similarly, the angle bisector of ∠B will divide it into two angles of y each.

**Hint**: The angle bisector divides an angle into two equal parts.

3. **Identify the Angles Formed by the Bisectors**:
- The angle bisectors will create a new quadrilateral, which we can denote as PQRS, where:
- ∠P = x (from bisector of ∠A)
- ∠Q = y (from bisector of ∠B)
- ∠R = x (from bisector of ∠C)
- ∠S = y (from bisector of ∠D)

**Hint**: Keep track of how the angles are divided by the bisectors.

4. **Analyze the Sum of Angles in Quadrilateral PQRS**:
- The sum of the angles in any quadrilateral is 360°. Therefore, we can write:
- ∠P + ∠Q + ∠R + ∠S = x + y + x + y = 2x + 2y = 360°.
- This implies that x + y = 90°.

**Hint**: Use the property that the sum of angles in a quadrilateral is always 360°.

5. **Determine the Type of Quadrilateral**:
- Since x + y = 90°, it indicates that each pair of opposite angles in quadrilateral PQRS are equal and each pair of adjacent angles are supplementary. This is the definition of a rectangle.
- Therefore, the quadrilateral formed by the angle bisectors of a parallelogram is a rectangle.

**Hint**: Recall that a rectangle has opposite angles equal and adjacent angles supplementary.

### Conclusion:
The bisectors of the angles of a parallelogram form a rectangle.