Euclid's elements book 3 proposition 20

[astrololo]

I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base."

(Figure is in the link) http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII20.html

English isn't my first language, so I just want to make sure that I understood something correctly. We prove the theorem by putting the two angles one on the other for the circumference. I was just wondering, can I assume that the angles do not need to be one on the other and they can have different portion of the circumference, as long as the circumference are of the same length ? (Will the proposition still work in this way?) I guess that Euclid did the proof by putting the angles one on the other for making the demonstration less wordy. (Less long to read)

Thank you!

geometry proof-verification euclidean-geometry

 

[HallsofIvy]

Yes, that is true. As long as you have the same circumference cut off you have the same angles.

 

[tommyxu3]

Do you mean the situation like below?
http://imageshack.com/a/img540/6139/5K0JNE.png

 

[tommyxu3]

If so, then it should be different, for the other angle is corresponding to the other arc.

 

[astrololo]

Yes, that is true. As long as you have the same circumference cut off you have the same angles.

I guess that I would also need to prove this then. right ?