Question

If a parallelogram is cyclic, then it is a _____

• A. rectangle
• B. square
• C. quadrilateral
• D. rhombus

Answer 1

The Correct Option is A

To solve the problem, we need to determine what type of parallelogram is cyclic. A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. Let us analyze the properties of a cyclic parallelogram.

Step 1: Properties of a Parallelogram A parallelogram has the following properties:

1. Opposite sides are parallel and equal in length.
2. Opposite angles are equal.
3. Consecutive angles are supplementary (i.e., they add up to \(180^\circ\)).

Step 2: Properties of a Cyclic Quadrilateral A cyclic quadrilateral has the following key property: The sum of the opposite angles is \(180^\circ\).

Step 3: Combining Properties For a parallelogram to be cyclic, it must satisfy both the properties of a parallelogram and the property of a cyclic quadrilateral. Specifically:
In a parallelogram, consecutive angles are supplementary.
For the parallelogram to be cyclic, opposite angles must also be supplementary. This implies that all angles of the parallelogram must be right angles (i.e., \(90^\circ\)). Why? Because if opposite angles are supplementary and consecutive angles are supplementary, the only way this can happen is if all angles are \(90^\circ\).
Step 4: Conclusion A parallelogram with all right angles is a rectangle. Therefore, if a parallelogram is cyclic, it must be a rectangle.

Final Answer: \[ {\text{rectangle}} \]