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# Cyclic Quadrilaterals

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Age 11 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

**Can you work out the angles in your triangles?**

Dan, Taylor, Grace and Summer from Long Field Spencer Academy in England, Ci Hui from Queensland College of Science and Technology in Australia, Thomas and Justin from Daegu International School in South Korea and Shaunak from Ganit Manthan, Vicharvatika in India worked out the angles in their triangles. Dan described a general method:

This is Ci Hui's work finding the angles in all of the possible triangles, using the same method:

**Can you work out the angles of your quadrilaterals?**

Freya, Summer, Katie and Phoebe from Long Field Spencer Academy, Ci Hui with images, Shaunak and Thomas and Justin worked out the angles in their quadrilaterals, and/or described a method for doing this. Freya wrote:

I created five quadrilaterals and calculated the angles by dividing them up into triangles. As I have already worked out all the angles in the triangles, I used that information to add up the angles in the quadrilaterals.

These are some of Katie's quadrilaterals, with the method shown (click on an image to open a larger version):

Some of Freya's quadrilaterals were different:

Shaunak had another different quadrilateral:

Finding the angles:

I drew 4 triangles in its inside. This is how it now looked:

I knew the angles of the four triangles. So, I added up the angles in each corner.

In the order of top left, top right, bottom right, bottom left, the angles are 80, 140, 100, 40.

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

James from Norwich School in the UK, Grace and Ci Hui noticed something about all of the angles. Grace wrote:

All of the angles together would have to add up to 360.

Ci Hui proved this. Click to see Ci Hui's work.

Katie, James, Ci Hui, Shaunak, Thomas and Justin and Freya noticed something about the opposite angles. Katie wrote:

The opposite angles add up to 180 degrees.

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

Thomas and Justin experimented with other numbers of dots:

We notice that the sum of the angles on opposite sides of the quadrilaterals is 180 degrees for all of the examples provided above.

**Extension:**

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

Sanaa from Heckmondwike Grammar School in England, James, Katie, Shaunak and Thomas and Justin thought the same thing would happen. Shaunak wrote:

I observed that the sum of opposite angles will be 180.

This must work for any number of dots because even if the number of dots change, the quadrilateral’s properties won’t change.

Sanaa, James and Thomas and Justin proved that the sum of opposite angles in a cyclic quadrilateral is always 180 degrees for cyclic quadrilaterals that contain the centre of the circle. This is Thomas and Justin's proof:

Thomas and Justin and Sanaa also proved that the opposite angles add up to 180 degrees even if the cyclic quadrilateral does not contain the center of the circle. This is Sanaa's proof (click on the image to open a larger version):

Ci Hui continued the investigation to see what would happen for polygons with other numbers of sides, and circles with other numbers of dots. Click to see the rest of Ci Hui's work.