How to Prove a Trapezium with Equal Non-Parallel Sides is Cyclic

Understanding the concept of a trapezium and a cyclic quadrilateral is crucial in geometry. A trapezium (or trapezoid) is a quadrilateral with exactly one pair of parallel sides, known as bases. A cyclic quadrilateral is a quadrilateral inscribed in a circle, meaning all its vertices lie on the circumference of the same circle. This article will explore how to prove that a trapezium with equal non-parallel sides is cyclic.

Understanding the Geometry

Let's consider a trapezium ABCD where AB and DC are the parallel sides, and AD and BC are the non-parallel sides which are equal in length. Importantly, if AB and DC were equal, the trapezium would transform into a parallelogram.

Step-by-Step Proof

Step 1: Establish Equal Base Angles

Given that AD and BC are equal in length, we can infer from the properties of isosceles triangles that the base angles DAB and ABC are equal. In other words:

∠DAB ∠ABC

Step 2: Use the Sum of Interior Angles Property

Since AB and DC are parallel, the sum of the interior angles on the same side of the transversal (line AB) is 180°. Therefore:

∠DAB ∠ADC 180°

and similarly,

∠ABC ∠BCD 180°

Step 3: Combine the Information

From the equality of the base angles and the sum of interior angles, we can deduce that:

∠ABC ∠ADC 180°

This statement is a characteristic property of a cyclic quadrilateral. It directly implies that the trapezium ABCD is inscribed in a circle, as opposite angles in a cyclic quadrilateral sum to 180°.

Conclusion

The proof demonstrates that if a trapezium has equal non-parallel sides (making it an isosceles trapezium), then its opposite angles sum to 180°. This characteristic is sufficient to conclude that the trapezium is cyclic. Therefore, trapezium ABCD is indeed a cyclic quadrilateral.

Frequently Asked Questions (FAQ)

Q1: Can all isosceles trapezia be cyclic?
Mainly, an isosceles trapezium with parallel bases (AB and DC in our case) can be cyclic, but it is not always the case for every segment of angles meeting the criteria.

Q2: What conditions must be met for a quadrilateral to be cyclic?
The key condition is that the opposite angles of the quadrilateral must sum to 180°. This is a fundamental criterion for cyclic quadrilaterals.

Q3: Is it possible for an irregular quadrilateral to have equal sides and be cyclic?
A quadrilateral with all equal sides (a rhombus) is irregular and cyclic as long as it fits the property of opposite angles summing to 180°.

References

For an in-depth study, one may refer to standard geometry texts and online resources such as the Wikipedia article on cyclic quadrilaterals.