SECTION 1.3 Circle Geometry 29
Proof. We draw a diameter, as
indicated; from the above lemma,
we see that θ 1 +ψ = 90. This
quickly leads toφ 1 = 2θ 1. Similarly
φ 2 = 2θ 2 , and we’re done.
Before proceeding, we shall find
the following concept useful. We
are given a circle and pointsA, B,
andP on the circle, as indicated
to the right. We shall say that the
angle APB̂ opens the arc ̆AB.
A degenerate instance of this is
when B and P agree, in which
case a tangent occurs. In this case
we shall continue to say that the given angleopensthe arcAB ̆.
As an immediate corollary to the Inscribed Angle Theorem, we get
the following:
Corollary. Two angles which
open the same are are equal.