Let ABCD is a parallelogram inscribed in circle.Since ABCD is a cyclic parallelogram, then∠A + ∠C = 180 ....1But ∠A = ∠CSo ∠A = ∠C = 90Again   ∠B + ∠D = 180 ....2But ∠B = ∠DSo ∠B = ∠D = 90Now each angle of parallelogram ABCD is 90.Hense ABCD is a rectangle.

Proof that a Cyclic Parallelogram is a Rectangle

A cyclic parallelogram is a special type of parallelogram that can be inscribed in a circle. In this proof, we will show that a cyclic parallelogram is indeed a rectangle.

Definition: A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

Definition: A rectangle is a parallelogram with all angles equal to 90 degrees.

Proof:

1. Consider a cyclic parallelogram ABCD inscribed in a circle.

2. Let's assume that ABCD is not a rectangle, which means that it does not have all angles equal to 90 degrees.

3. Since ABCD is a parallelogram, opposite angles are equal. Let's assume that angle A is not equal to 90 degrees.

4. Since ABCD is a cyclic parallelogram, opposite angles are supplementary (they add up to 180 degrees) because they are inscribed in the same arc.

5. Therefore, angle A + angle C = 180 degrees.

6. Since ABCD is a parallelogram, opposite sides are parallel. Let's assume that AB and CD are parallel.

7. By the property of alternate interior angles, angle A is equal to angle C.

8. Substituting angle A with angle C in the equation from step 5, we get 2 * angle C = 180 degrees.

9. Solving for angle C, we find that angle C = 90 degrees.

10. However, this contradicts our assumption in step 3 that angle A is not equal to 90 degrees.

11. Hence, our assumption that ABCD is not a rectangle must be incorrect.

12. Therefore, a cyclic parallelogram ABCD is a rectangle.

Conclusion: We have proved that a cyclic parallelogram is a rectangle by contradiction. By assuming that it is not a rectangle, we arrived at a contradiction, which proves that our assumption was incorrect. Therefore, a cyclic parallelogram must be a rectangle.