Prove that any non-isosceles trapezium is not cyclic.

To demonstrate that any non-isosceles trapezium cannot be cyclic, we need to delve into some properties of trapeziums and cyclic quadrilaterals. A cyclic quadrilateral is one where all four vertices lie on the circumference of a single circle. One of the key properties of cyclic quadrilaterals is that the opposite angles sum up to 180 degrees. Let's break this down step by step.

Understanding the Properties of Trapeziums

A trapezium (or trapezoid in some regions) is defined as a quadrilateral with at least one pair of parallel sides. In the case of a non-isosceles trapezium, the lengths of the non-parallel sides (the legs) are not equal. This asymmetry plays a crucial role in our proof.

Angles in a Trapezium

In a trapezium, the angles adjacent to each base are supplementary. This means that if we denote the angles at the top base as A and B, and the angles at the bottom base as C and D, we have:

• A + D = 180 degrees
• B + C = 180 degrees

However, in a non-isosceles trapezium, the angles A and B are not equal to angles C and D, respectively. This is due to the unequal lengths of the legs, which causes the angles to differ.

Exploring the Cyclic Condition

For a quadrilateral to be cyclic, the sum of the opposite angles must equal 180 degrees. In our case, we need to check if:

• A + C = 180 degrees
• B + D = 180 degrees

However, since A and B are not equal to C and D, we cannot guarantee that A + C will equal 180 degrees. This inequality arises from the fact that the legs of a non-isosceles trapezium create angles that are not symmetrical.

Illustrative Example

Consider a non-isosceles trapezium ABCD, where AB is parallel to CD. Let’s say AB is shorter than CD. The angles A and B will be acute, while angles C and D will be obtuse. If we try to calculate A + C, we find that it does not equal 180 degrees due to the differing lengths of the legs AD and BC. This discrepancy confirms that the opposite angles do not satisfy the cyclic condition.

Conclusion of the Proof

Since we have established that in a non-isosceles trapezium, the opposite angles do not sum to 180 degrees, it follows that such a trapezium cannot be inscribed in a circle. Therefore, we conclude that any non-isosceles trapezium is not cyclic. This property is essential in understanding the geometric relationships within quadrilaterals and highlights the unique characteristics of isosceles trapeziums, which do meet the cyclic condition due to their symmetry.