Opposite Sides of a Circumscribing Quadrilateral Subtend Supplementary Angles at the Centre of the Circle

To prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle, we can use the concept of cyclic quadrilaterals and properties of angles in a circle.

Cyclic Quadrilaterals:
A quadrilateral is called cyclic if all its four vertices lie on a circle. In other words, if a quadrilateral can be inscribed within a circle, it is a cyclic quadrilateral.

Properties of Cyclic Quadrilaterals:
1. Opposite angles in a cyclic quadrilateral are supplementary.
2. The sum of the opposite angles of a cyclic quadrilateral is 180 degrees.

Proof:

Step 1:
Consider a quadrilateral ABCD that is circumscribed around a circle with center O.

Step 2:
Let the opposite sides of the quadrilateral be AB and CD, and let them intersect at point P.

Step 3:
Since ABCD is circumscribed around the circle, the quadrilateral is a cyclic quadrilateral.

Step 4:
According to the properties of cyclic quadrilaterals, the opposite angles in the quadrilateral are supplementary. Therefore, angle APD + angle BPC = 180 degrees.

Step 5:
Now, let's consider the angles subtended by AB and CD at the center of the circle, which is point O. Let these angles be α and β, respectively.

Step 6:
As angle APD subtends α at the center, it also subtends α at any other point on the circumference of the circle.

Step 7:
Similarly, as angle BPC subtends β at the center, it also subtends β at any other point on the circumference of the circle.

Step 8:
Since the opposite angles in a cyclic quadrilateral are supplementary (angle APD + angle BPC = 180 degrees), α + β = 180 degrees.

Step 9:
This proves that the opposite sides of the quadrilateral ABCD subtend supplementary angles α and β at the center of the circle.

Conclusion:
Hence, it has been proved that in a quadrilateral circumscribing a circle, the opposite sides subtend supplementary angles at the center of the circle.