Consider the following statements about special quadrilaterals: (1) Every isosceles trapezium is cyclic. (2) Any parallelogram that is cyclic must be a rectangle. Which of the above statements is or are correct?

Introduction / Context:
This is a conceptual geometry question about cyclic quadrilaterals and special quadrilaterals like trapeziums and parallelograms. It asks you to judge the truth of two statements involving isosceles trapeziums and cyclic parallelograms. Understanding properties of cyclic quadrilaterals and angle relationships is essential here.

Given Data / Assumptions:

• Statement 1: Every isosceles trapezium is cyclic.
• Statement 2: Any parallelogram that is cyclic must be a rectangle.
• A trapezium (trapezoid) has exactly one pair of parallel sides.
• An isosceles trapezium is a trapezium whose non parallel sides (legs) are equal in length.
• A quadrilateral is cyclic if all its vertices lie on a single circle.
• For a cyclic quadrilateral, opposite angles are supplementary (their sum is 180 degrees).

Concept / Approach:
We analyse each statement separately using known theorems:

• For a trapezium to be cyclic, a necessary and sufficient condition is that its non parallel sides are equal, that is, the trapezium is isosceles.
• For a parallelogram to be cyclic, opposite angles must be supplementary. However, in a parallelogram, opposite angles are already equal. The only way for an angle to be equal to its supplement is for it to be 90 degrees, which makes the parallelogram a rectangle.

These facts directly address each statement.

Step-by-Step Solution:
Step 1: Consider Statement 1: Every isosceles trapezium is cyclic. Step 2: In an isosceles trapezium, the legs are equal and the base angles at each base are equal. The sum of a pair of adjacent interior angles between a leg and a base is 180 degrees. Step 3: If a quadrilateral has one pair of opposite angles supplementary, and the quadrilateral is a trapezium with equal legs, it can be inscribed in a circle. In fact, a trapezium is cyclic if and only if it is isosceles. Thus every isosceles trapezium is cyclic, so Statement 1 is correct. Step 4: Now consider Statement 2: Any cyclic parallelogram is a rectangle. Step 5: In a parallelogram, opposite angles are equal. In a cyclic quadrilateral, opposite angles are supplementary (sum is 180 degrees). Step 6: Combine these: if a parallelogram is cyclic, then each pair of opposite equal angles must also be supplementary. This is only possible if each of those angles is 90 degrees. Step 7: A parallelogram with all right angles is by definition a rectangle. Hence any cyclic parallelogram must be a rectangle, so Statement 2 is also correct. Step 8: Since both statements are correct, the correct choice is that both 1 and 2 are true.

Verification / Alternative check:
You can draw an isosceles trapezium and see that its vertices lie on a circle by constructing equal arcs and using base angle equality. For the parallelogram case, try drawing a non right angled parallelogram and attempt to inscribe it in a circle: the sum of opposite angles will exceed or fall short of 180 degrees, making a circumcircle impossible. Only a rectangle satisfies both parallelogram and cyclic conditions at once.

Why Other Options Are Wrong:
1 only or 2 only: Each would assert that exactly one statement is true. We have shown that both statements 1 and 2 are correct, so these options are invalid.
Neither 1 nor 2: This would mean both statements are false, which directly contradicts established theorems.
Cannot be determined: The truth of these statements comes from standard Euclidean geometry and does not require additional numerical data, so it can be determined.

Common Pitfalls:
One common misconception is to think that any trapezium can be cyclic, which is not correct. Only isosceles trapeziums guarantee cyclicity. Another mistake is to forget that a cyclic quadrilateral has supplementary opposite angles and not equal opposite angles in general. Confusing these properties may lead to wrong conclusions about parallelograms. Remembering that cyclicity and parallelogram properties together force right angles helps avoid error.

Final Answer:
Both statements are correct, so the right choice is Both 1 and 2.