2. If two chords intersect within a circle, the angle which they include is measured by half the sum of the in- tercepted arcs.
Let the chords AB and CD of a circle intersect at E; join AD.
The angle AEC is equal to the angles ADE, and DAE, by I. 32; that is, to the angles standing on the arcs AC and BD. Thus the angle AEC is equal to an angle at the circumference of the circle standing on the sum of the arcs AC and BD; and is therefore equal to an angle at the centre of the circle standing on half the sum of these arcs.
Similarly the angle CEB is measured by half the sum of the arcs CB and AD.
3. If two chords produced intersect without a circle, the angle which they include is measured by half the difference of the intercepted arcs.
Let the chords AB and CD of a circle, produced, intersect at E; join AD.
The angle ADC is equal to the angles EAD and AED, by I. 32. Thus the angle AEC is equal to the difference of the angles ADC and BAD; that is, to an angle at the circumference of the circle standing on an arc which is the
difference of AC and BD; and is therefore equal to an angle at the centre of the circle standing on half the difference of these arcs.
