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# Cyclic Quadrilaterals Proof

*You may be interested in the other problems in our Sharpen your Skills Feature.*

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*

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Age 11 to 16

Challenge Level

*This problem follows on from Cyclic Quadrilaterals*

Sketch a circle and choose four points at random to form a quadrilateral.

Can you prove that the opposite angles of your quadrilateral add to $180^\circ$?

Click below to see a diagram that might help you to prove it.

*Quadrilaterals whose vertices lie on the edge of a circle are called Cyclic Quadrilaterals.*

**Extension**

Can you prove that when the centre of the circle is outside the cyclic quadrilateral, the opposite angles also add to $180^\circ$?

You may wish to draw some examples on

9, 10, 12, 15 and 18 dot circles.

Click below to see a diagram that might help you to prove that the opposite angles of cyclic quadrilaterals add to $180^\circ$, when the centre of the circle is outside the cyclic quadrilateral.