1. **Problem Statement:** Prove that a cyclic parallelogram is a rectangle. 2. **Key Definitions:** - A parallelogram is a quadrilateral with opposite sides parallel. - A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, meaning all its vertices lie on the circle. - A rectangle is a parallelogram with all angles equal to 90 degrees. 3. **Formula and Properties Used:** - Opposite angles of a parallelogram are equal. - The sum of opposite angles in a cyclic quadrilateral is 180 degrees. 4. **Proof Steps:** - Let the parallelogram be ABCD, with vertices on the circle. - Since ABCD is cyclic, the sum of opposite angles is 180 degrees: $$\angle A + \angle C = 180^\circ$$ - Since ABCD is a parallelogram, opposite angles are equal: $$\angle A = \angle C$$ - From the two equations, we have $$\angle A + \angle A = 180^\circ \Rightarrow 2\angle A = 180^\circ \Rightarrow \angle A = 90^\circ$$ - Therefore, all angles are 90 degrees (since opposite angles are equal and adjacent angles in a parallelogram sum to 180 degrees). 5. **Conclusion:** - Since all angles are right angles, the cyclic parallelogram ABCD is a rectangle. **Final answer:** A cyclic parallelogram must be a rectangle because its angles are all right angles.