Arcs Intercepted by Secants

In the following figure, PT and QT are secants to circle O.  The measure of angle T is related to the measures of arcs PQ and RS by the given formula:

The proof of this is similar to the proof that the measure of the angle formed by two intersecting chords is the average of intercepted arcs.

In triangle QRT, angle PRQ is an exterior angle, so it is equal to the sum of its remote interior angles:

But angle PRQ is an inscribed angle, so its measure is half of its intercepted arc, of half of a.  Likewise, angle RQT is also an inscribed angle, so it measures half of b.  Substituting these in the above equation gives us:

Therefore,

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