Introduction
In geometry, a trapezium (or trapezoid) with equal non-parallel sides is known as an isosceles trapezium. We aim to prove that such a trapezium is cyclic, meaning that all its vertices lie on a single circle.
Definition of Cyclic Quadrilateral
A quadrilateral is cyclic if the sum of each pair of opposite angles is equal to 180 degrees.
Properties of Isosceles Trapezium
- Let trapezium ABCD have AB || CD and AD = BC (the non-parallel sides).
- Angles at the bases:
- Angle A + Angle B = 180 degrees
- Angle C + Angle D = 180 degrees
Proof Steps
1. Base Angles:
- Since AB || CD, by the Alternate Interior Angle Theorem:
- Angle A = Angle D
- Angle B = Angle C
2. Sum of Angles:
- Adding the angles results in:
- Angle A + Angle B + Angle C + Angle D = 360 degrees
3. Using Equal Sides:
- Since AD = BC, the base angles (Angle A and Angle B) are equal:
- Angle A + Angle B = Angle C + Angle D
4. Establishing Cyclic Condition:
- As shown:
- Angle A + Angle D = 180 degrees (because they are equal)
- Angle B + Angle C = 180 degrees (because they are equal)
Conclusion
Since the sum of opposite angles is 180 degrees, we conclude that trapezium ABCD is cyclic. Thus, any isosceles trapezium can be inscribed in a circle, confirming that it is a cyclic quadrilateral.