How are the angles in the chordal quadrilateral problem related?

[tomkoolen]

Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:
"Prove that ABFE is a chordal quadrilateral." (see attachment)


Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90o- 1/2*arc ED.
Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90o- 1/2*arc ED.
=> Angle A = Angle EFC.
Angle A + Angle EFB = Angle EFC + Angle EFB = 180o.
Thus ABFE is a chordal quadrilateral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Thanks in advance,
Tom

 

[tiny-tim]

Hello Tom!

You need to know the theorem that the angle subtended by an arc at any point on the rest of the circumference is half the angle subtended by that arc at the centre (ie half the arc).