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

Charlie drew a circle and chose four points at random to form a quadrilateral.

Can you prove that the opposite angles of his quadrilateral also 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.*

So far you have only considered what happens to the opposite angles of cyclic quadrilaterals which contain the centre of the circle inside them...

**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 $180^\circ$.

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

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 $180^\circ$.