# Prove that the oppsite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre

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by mazzini

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by mazzini

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join AO,BO,CO,DO.

The angle BAo,DAO is equal as AB and AD are tanglents

let each angle equal = a;

The angles at B are similarly equal to each other. Let each of them equal b.

Similarly for vertices C and D.

The sum of the angles at the centre is 360 deg.

The sum of the angles of ABCD is 360 deg.

Therefore:

2(a + b + c + d) = 360

a + b + c + d = 180.

From triangle AOB, angle BOA = 180 - (a + b).

From triangle COD, angle COD = 180 - (c + d).

Angle BOA + angle COD = 360 - (a + b + c + d)

= 360 - 180

= 180 deg.

Thus AB and CD subtend supplementary angles at O.