Okay, circles is my least favourite part of maths, I find it really hard, and a lot of the questions I just don't get. Here is one of them, please can you explain how to work it out?:

12:
ABCDEF is a hexagon inscribed in a circle. By joining AD, prove that angle ABC + angle CDE +angle EFA = 360°

How do I do this? I don't understand, how do I prove it just from this information?

Please help!
Thanks,
Lucy

You don't need the hexagon to be regular. It can be any hexagon as long as it's six point lie on the circle.

Step One
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Draw a circle. Put six point on the circumference. Without loss of generality, spread them out, so you can see what's going on. Join AC, EC and EA.

Note you have a triangle inside the hexagon.

Step Two
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Add the angles of the triangle.
AEC + ECA + CAE = 180o

Step three.
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Find EDC in terms of CAE.
This is the nasty part of the question, or the A HAH moment. On another sheet of paper draw a circle. Put one line in the circle. Don't make this the diameter.

Label this line EC.

Now pick a point on the circle that is on one side of EC. Call it A
Pick another point on the opposite side of EC. Call it D.

Do you know that angles EAC + EDC = 180o? If you don't my proof will not work.

EDC = 180 - EAC.

Step Four
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Find the other two relationships.
AFE = 180 - ACE
ABC = !80 - AED

Step Five
Add your three angles together.
ABC + AFE + EDC = 180 - AED + 180 - ACE + 180 - ECE
ABC + AFE + EDC = 540 - AED - ACE - ECE
ABC + AFE + EDC = 540 - (180) Why??? Hint: look at the triangle.
360