Introduction:
A parallelogram is a quadrilateral with opposite sides parallel and equal in length. In this proof, we will show that the bisectors of the interior angles of a parallelogram enclose a rectangle.

Proof:
Let's consider a parallelogram ABCD, where AB is parallel to CD and AD is parallel to BC.

Claim 1: The bisectors of the interior angles of a parallelogram are concurrent.
Proof of Claim 1:
To prove this claim, we can use the fact that opposite angles in a parallelogram are equal. Let the bisectors of angle A and angle C intersect at point E, and the bisectors of angle B and angle D intersect at point F.

- By the definition of an angle bisector, angle EAD = angle EAB and angle ECD = angle ECB.
- Since AB is parallel to CD, angle EAB = angle ECB (opposite angles in a parallelogram are equal).
- Therefore, angle EAD = angle ECD.

Similarly, we can prove that angle FAB = angle FCB.

- By the definition of an angle bisector, angle FAB = angle FAD and angle FCB = angle FCD.
- Since AD is parallel to BC, angle FAD = angle FCD (opposite angles in a parallelogram are equal).
- Therefore, angle FAB = angle FCB.

Thus, we have shown that the bisectors of the interior angles of a parallelogram are concurrent at point E and point F.

Claim 2: The bisectors of the interior angles of a parallelogram are perpendicular to each other.
Proof of Claim 2:
To prove this claim, we can use the fact that opposite sides in a parallelogram are equal and parallel.

- Let's consider the bisectors of angle A and angle C. These bisectors intersect at point E.
- By the definition of an angle bisector, angle EAB = angle EAD and angle ECD = angle ECB.
- Since AB is parallel to CD, angle EAB = angle ECB (opposite angles in a parallelogram are equal).
- Therefore, angle EAD = angle ECD.

Since angle EAD and angle ECD are equal, and opposite sides in a parallelogram are equal, we can conclude that triangle AED is an isosceles triangle.

Similarly, we can prove that triangle BEC is an isosceles triangle.

- Let's consider the bisectors of angle B and angle D. These bisectors intersect at point F.
- By the definition of an angle bisector, angle FAB = angle FAD and angle FCB = angle FCD.
- Since AD is parallel to BC, angle FAD = angle FCD (opposite angles in a parallelogram are equal).
- Therefore, angle FAB = angle FCB.

Since angle FAB and angle FCB are equal, and opposite sides in a parallelogram are equal, we can conclude that triangle ABF is an isosceles triangle.

Conclusion:
From the above claims, we can conclude that the bisectors of the interior angles of a parallelogram enclose a rectangle. The intersection points E and F are the opposite vertices of the rectangle, and the sides of the rectangle