Question

Prove that the bisectors of the angles of a parallelogram form a rectangle.

 

Answer 1

Given: Parallelogram PQRS.

To Prove: The quadrilateral formed by the bisectors of the angles of parallelogram PQRS is a rectangle.

Proof:

1. Let the bisectors of the angles at vertices P, Q, R and S intersect in such a way that they form a quadrilateral inside the parallelogram. Name the intersection points of the adjacent bisectors as A, B, C and D respectively (as shown in the figure).

2. Since PQRS is a parallelogram, opposite angles are equal: ∠P = ∠R and ∠Q = ∠S

3. The sum of angles at adjacent vertices of a parallelogram is 180°: ∠P + ∠Q = 180°

4. The bisector of an angle divides it into two equal parts. Therefore, the angle between two adjacent bisectors (such as those at P and Q) inside the quadrilateral is half the sum of adjacent angles of the parallelogram:

∠Between adjacent bisectors = `(∠P)/2 + (∠Q)/2`

= `(∠P + ∠Q)/2`

= `180^circ/2`

= 90°

5. Thus, each interior angle of the quadrilateral formed by the angle bisectors is 90°, making it a rectangle.

The quadrilateral formed by bisectors of the angles of a parallelogram is a rectangle, because all its interior angles are right angles.