Challenge Level

*If you prefer to work on paper, you may wish to print out some sheets of circles:
9 dot 10 dot 12 dot 15 dot 18 dot*

In the GeoGebra interactivity below there is

Can you work out the angles in your triangles?

You should have found four different triangles with angles of:

40, 70, 70

80, 50, 50

120, 30, 30,

160, 10, 10

80, 50, 50

120, 30, 30,

160, 10, 10

**Now draw a few quadrilaterals whose interior contains the centre of the circle, by joining four dots on the edge.**

Can you work out the angles of your quadrilaterals?

*If you're finding it hard to work out the angles, take a look at Getting Started.*

Create at least five different quadrilaterals in this way and work out their angles.

**What do you notice about the angles on opposite vertices of your quadrilaterals?**

**Perhaps you are wondering whether this only happens with 9-dot circles...**

You may wish to explore the opposite angles of quadrilaterals on circles with a different number of dots.

Click below for interactivities with 10, 12, 15 and 18 dots around the circle.

10 dots

12 dots

15 dots

18 dots

**Extension:**

**Will the same happen if you draw a circle and choose four points at random to form a quadrilateral?**

To prove that the opposite angles of all cyclic quadrilaterals add to $180^\circ$ go to Cyclic Quadrilaterals Proof

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