PROPOSITION 27. THEOREM.
In equal circles, the angles which stand on equal arcs are equal to one another whether they he at the centres or circumferences.
Let ABC, DEF be equal circles, and let the angles BGC, EHF at their centres, and the angles BAC, EDF at their circumferences, stand on equal arcs BC, EF: the angle BGC shall be equal to the angle EHF, and the angle BAC equal to the angle EDF.
If the angle BGC be equal to the angle EHF, it is manifest that the angle BAC is also equal to the angle EDF. [III. 20, Axiom 7.
But, if not, one of them must be the greater. Let BGC be the greater, and at the point G, in the straight line BG, make the angle BGK equal to the angle EHF. [I. 23.
Then, because the angle BGK is equal to the angle EHF, and that in equal circles equal angles stand on equal arcs, when they are at the centres, [III. 26.
therefore the arc BK is equal to the arc EF.
But the arc EF is equal to the arc BC; [Hypothesis.
therefore the arc BK is equal to the arc BC, [Axiom 1.
the less to the greater; which is impossible.
Therefore the angle BGC is not unequal to the angle EHF, that is, it is equal to it.
And the angle at A is half of the angle BGC, and the angle at D is half of the angle EHF; [III. 20.
therefore the angle at A is equal to the angle at D. [Ax. 7.
Wherefore, in equal circles &c. q.e.d.
